NUMERICAL METHODS. Numerical Methods SCE 1 CIVIL ENGINEERING

Transcription

1 Nmerical Methods SCE CIVIL ENGINEERING

2 MA659 L T P C OBJECTIVES: This corse aims at providing the necessar basic concepts of a few nmerical methods and give procedres for solving nmericall different kinds of problems occrring in engineering and technolog UNIT I SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS + Soltion of algebraic and transcendental eqations - Fied point iteration method Newton Raphson method- Soltion of linear sstem of eqations - Gass elimination method Pivoting - Gass Jordan method Iterative methods of Gass Jacobi and Gass Seidel - Matri Inversion b Gass Jordan method - Eigen vales of a matri b Power method. UNIT II INTERPOLATION AND APPROXIMATION 8+ Interpolation with neqal intervals - Lagrange's interpolation Newton s divided difference Interpolation Cbic Splines - Interpolation with eqal intervals - Newton s forward and backward difference formlae. UNIT III NUMERICAL DIFFERENTIATION AND INTEGRATION 9+ Approimation of derivatives sing interpolation polnomials - Nmerical integration sing Trapezoidal Simpson s / rle Romberg s method - Two point and three point Gassian Qadratre formlae Evalation of doble integrals b Trapezoidal and Simpson s / rles. UNIT IV INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIALEQUATIONS 9+ Single Step methods - Talor s series method - Eler s method - Modified Eler s method Forth order Rnge-Ktta method for solving first order eqations - Mlti step methods - Milne s and Adams-Bash forth predictor corrector methods for solving first order eqations. UNIT V BOUNDARY VALUE PROBLEMS IN ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS 9+ Finite difference methods for solving two-point linear bondar vale problems - Finite difference Techniqes for the soltion of two dimensional Laplace s and Poisson s eqations on rectanglar Domain One dimensional heat flow eqation b eplicit and implicit Crank Nicholson methods One dimensional wave eqation b eplicit method. TOTAL L:5+T:5: 6 PERIODS OUTCOMES: The stdents will have a clear perception of the power of nmerical techniqes ideas and wold be able to demonstrate the applications of these techniqes to problems drawn from indstr management and other engineering fields. TEXT BOOKS:. Grewal. B.S. and Grewal. J.S."Nmerical methods in Engineering and Science" Khanna Pblishers 9th Edition New Delhi 7.. Gerald. C. F. and Wheatle. P. O. "Applied Nmerical Analsis" Pearson Edcation Asia 6th Edition New Delhi 6. REFERENCES:. Chapra. S.C. and Canale.R.P. "Nmerical Methods for Engineers Tata McGraw Hill 5th Edition New Delhi 7. Brian Bradie. "A friendl introdction to Nmerical analsis" Pearson Edcation AsiaNew Delhi 7.. Sankara Rao. K. "Nmerical methods for Scientists and Engineers" Prentice Hall of IndiaPrivate rd Edition New Delhi 7. Contents SCE CIVIL ENGINEERING

3 UNIT I Soltion of Eqations & Eigen Vale Problems 7. Nmerical soltion of Non-Linear Eqations 7 Method of false position 8 Newton Raphson Method 9 Iteration Method. Sstem of Linear Eqations Gass Elimination Method Gass Jordan Method 5 Gass Jacobi Method 7 Gass Seidel Method 9. Matri Inversion Inversion b Gass Jordan Method. Eigen Vale of a Matri Von Mise`s power method Ttorial Problems Qestion Bank 6 UNIT- II Interpolation and approimation. Interpolation with Uneqal Intervals Lagrange`s Interpolation formla Inverse Interpolation b Lagrange`s Interpolation Polnomial. Divided Differences-Newton Divided Difference Interpolation Divided Differences Newtons Divided Difference formla for neqal intervals 5. Interpolating with a cbic spline 7 Cbic spline interpolation 8. Interpolation with eqals SCE CIVIL ENGINEERING

4 Newtons` forward interpolation formla Newtons` Backward interpolation formla Ttorial Problems Qestion Bank 6 UNIT- III Nmerical Differentiation and Integration 5. Nmerical Differentiation Derivatives sing divided differences 5 Derivatives sing finite Differences 5 Newton`s forward interpolation formla 5 Newton`s Backward interpolation formla 5. Nmerical integration 5 Trapezoidal Rle 5 Simpson`s / Rle 5 Simpson`s /8 Rle 55 Romberg`s intergration 56. Gassian qadratre 58 Two Point Gassian formla & Three Point Gassian formla 59. Doble integrals 6 Trapezoidal Rle & Simpson`s Rle 6 Ttorial Problems 65 Qestion Bank 66 UNIT IV Nmerical soltion of Ordinar Differential Eqation 67. Talor`s Series Method 67 Power Series Soltion 67 Pointwise Soltion 68 Talor series method for Simltaneos first order Differential eqations 7 SCE 5 CIVIL ENGINEERING

5 . Eler Methods 7 Eler Method 7 Modified Eler Method 7. Rnge Ktta Method 7 Forth order Rnge-ktta Method 75 Rnge-Ktta Method for second order differential eqations 76. Mlti-step Methods 78 Milne`s Method 79 Adam`s Method 8 Ttorial Problems 8 UNIT V Bondar Vale Problems in ODE & PDE Soltion of Bondar Vale Problems in ODE Soltion of Laplace Eqation and Poisson Eqation 9 Soltion of Laplace Eqation Leibmann`s iteration process 9 Soltion of Poisson Eqation 9 5. Soltion of One Dimensional Heat Eqation 95 Bender-Schmidt Method 96 Crank- Nicholson Method Soltion of One Dimensional Wave Eqation 99 Ttorial Problems Qestion Bank 6 Previos ear Universit Qestion paper 7 SCE 6 CIVIL ENGINEERING

6 CHAPTER - SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. Nmerical soltion of Non-Linear eqations Introdction The problem of solving the eqation is of great importance in science and Engineering. In this section we deal with the varios methods which give a soltion for the eqation Soltion of Algebraic and transcendental eqations The eqation of the form f = are called algebraic eqations if f is prel a polnomial in.for eample: are algebraic eqations. If f also contains trigonometric logarithmic eponential fnction etc. then the eqation is known as transcendental eqation. Methods for solving the eqation The following reslt helps s to locate the interval in which the roots of Method of false position. Iteration method Newton-Raphson method Method of False position Or Regla-Falsi method Or Linear interpolation method In bisection method the interval is alwas divided into half. If a fnction changes sign over an interval the fnction vale at the mid-point is evalated. In bisection method the interval from a to b into eqal intervals no accont is taken of the magnitde of.an alternative method that eploits this graphical insights is to join b a straight line.the intersection of this line with the X-ais represents an improved estimate of the root.the replacement of the crve b a straight line gives a false position of the root is the origin of the name method of false position or in Latin Regla falsi.it is also called the linear interpolation method. Problems. Find a real root of that lies between and b the method of false position and correct to three decimal places. Let The root lies between.5 and. The approimations are given b af b bf a f b f a Iterationr a b r f r SCE 7 CIVIL ENGINEERING

7 Obtain the real root of correct to for decimal places sing the method of false position. Given Taking logarithmic on both sides log f log f and f.6.7 The roots lies between.5 and.6 The approimation are given b af b f b bf a f a Iterationr a b r f r The reqired root is.597 Eercise:. Determine the real root of correct to for decimal places b Regla-Falsi method. Ans:.99. Find the positive real root of correct to for decimals b the method of False position. Ans:.8955.Solve the eqation Ans:.798 b Regla-Falsi methodcorrect to decimal places. Newton s method or Newton-Raphson method Or Method of tangents SCE 8 CIVIL ENGINEERING

8 This method starts with an initial approimation to the root of an eqation a better and closer approimation to the root can be fond b sing an iterative process. Derivation of Newton-Raphson formla Let be the root of f and be an approimation to.if h Then b the Taor s series i i f i i= f i Note: The error at an stage is proportional to the sqare of the error in the previos stage. The order of convergence of the Newton-Raphson method is at least or the convergence of N.R method is Qadratic. Problems.Using Newton s method find the root between and of places. correct five decimal Given f 6 f 6 B Newton-Raphson formla we have approimation The initial approimation is. 5 i f i f i n i... First approimation: f f.5 =.79 Second approimation: SCE 9 CIVIL ENGINEERING

9 f f.79 = Third approimation f f Forth approimation.75 =.75 f f =.75 The root is.75 correct to five decimal places.. Find a real root of =/+sin near.5 correct to decimal places b newton-rapson method. f sin Let f cos B Newton-Raphson formla we have approimation i f i f i n i... n sin n.5 n cos n n cos Let. 5be the initial approimation n... First approimation: SCE CIVIL ENGINEERING

10 .5.5 sin.5 cos =.97 Second approimation: =.97. The reqired root is.97 Eercise:. Find the real root of e that lies between and b Newton s method correct to decimal places. Ans:.5.Use Newton-Raphson method to solve the eqation cos Ans:.67.Find the doble root of the eqation b Newton s method. Ans:.6 Iteration method Or Method of sccessive approimationsorfied point method For solving the eqation f b iteration method we start with an approimation vale of the root.the eqation f is epressed as.the eqation is called fied point eqation.the iteration formla is given b n n n... called fied point iteration formla. TheoremFied point theorem Let f be the given eqation whose eact root is. The eqation f be rewritten as.let I be the interval containing the root. If for all in I then the seqence of Approimation... n will converges to. If the initial starting vale is chosen in I. SCE CIVIL ENGINEERING

11 The order of convergence Theorem Let. be a root of the eqation g.if g g p... g p andg.then the convergence of iteration g is of order p. i i Note The order of convergence in general is linear i.e = Problems.Solve the eqation or the positive root b iteration method correct to for decimal places. f f andf The root lies between and. The given eqation can be epressed as and Choosing. 75 the sccessive approimations are SCE CIVIL ENGINEERING

12 .75 =.7559 Hence the root is Eercise:. Find the cbe root of 5 correct to for decimal places b iteration method Ans:.66. Sstem of linear eqation Introdction Man problems in Engineering and science needs the soltion of a sstem of simltaneos linear eqations.the soltion of a sstem of simltaneos linear eqations is obtained b the following two tpes of methods Direct methods Gass elimination and Gass Jordan method Indirect methods or iterative methods Gass Jacobi and Gass Seidel method a Direct methods are those in which The comptation can be completed in a finite nmber of steps reslting in the eact soltion The amont of comptation involved can be specified in advance. The method is independent of the accrac desired. b Iterative methods self correcting methods are which Begin with an approimate soltion and Obtain an improved soltion with each step of iteration Bt wold reqire an infinite nmber of steps to obtain an eact soltion withot rond-off errors The accrac of the soltion depends on the nmber of iterations performed. SCE CIVIL ENGINEERING

13 Simltaneos linear eqations The sstem of eqations in n nknowns... n is given b a a a... a n n b a... an n b.. a a... n n a mn n b n This can be written as AX B where T ij ; X n n... n ; B b b b A a... b n T This sstem of eqations can be solved b sing determinants Cramer s rle or b means of matrices. These involve tedios calclations. These are other methods to solve sch eqations.in this chapter we will discss for methods viz. i Gass Elimination method ii Gass Jordan method iii Gass-Jacobi method iv Gass seidel method Gass-Elimination method This is an Elimination method and it redces the given sstem of eqation to an eqivalent pper trianglar sstem which can be solved b Back sbstittion. Consider the sstem of eqations a a a a a a a a a b b b Gass-algorithm is eplained below: Step. Elimination of from the second and third eqations.if a the first eqation is sed to eliminate from the second and third eqation. After elimination the redced sstem is SCE CIVIL ENGINEERING

14 a a a a a a a b b b Step : Elimination of from the third eqation. If a We eliminate from third eqation and the redced pper trianglar sstem is a a a a a b a b b Step : From third eqation is known. Using in the second eqation is obtained. sing both And in the first eqation the vale of is obtained. Ths the elimination method we start with the agmented matri A/B of the given sstem and transform it to U/K b eliminator row operations. Finall the soltion is obtained b back sbstittion process. Principle A/ B Gass e lim ination U / K. Gass Jordan method This method is a modification of Gass-Elimination method. Here the elimination of nknowns is performed not onl in the eqations below bt also in the eqations above. The co-efficient matri A of the sstem AX=B is redced into a diagonal or a nit matri and the soltion is obtained directl withot back sbstittion process. Gass Jordan A/ B D / K or I / K Eamples.Solve the eqations z 8 z z b Gass Elimination method Gass Jordan method SCE 5 CIVIL ENGINEERING

15 igass-elimination method The given sstem is eqivalent to AX B where and Principle. Redce to = From this the eqivalent pper trianglar sstem of eqations is z 7 z 7z 8 7 z=== B back sbstittion. The soltion is ==z= iigass Jordan method Principle. Redce to SCE 6 CIVIL ENGINEERING

16 From this we have = z= Eercise. Solve the sstem b Gass Elimination method5 z z z 5. Ans: = 9.5;=6.79;z= Solve the eqations z 9 z 5z b Gass Elimination method Gass Jordan method Ans: =; =; z=5 Iterative method These methods are sed to solve a special of linear eqations in which each eqation mst possess one large coefficient and the large coefficient mst be attached to a different nknown in that eqation.frther in each eqation the absolte vale of the large coefficient of the nknown is greater than the sm of the absolte vales of the other coefficients of the other nknowns. Sch tpe of simltaneos linear eqations can be solved b the following iterative methods. Gass-Jacobi method Gass seidel method SCE 7 CIVIL ENGINEERING

17 SCE 8 CIVIL ENGINEERING d z c b a d z c b a d z c b a i.e. the co-efficient matri is diagonall dominant. Solving the given sstem for z whose diagonals are the largest valeswe have ] [ ] [ ] [ b a d c z z c a d b z c b d a Gass-Jacobi method If the r th iterates are r r r z then the iteration scheme for this method is r r r r r r r r r b a d c z z c a d b z c b d a The iteration is stopped when the vales z start repeating with the desired degree of accrac. Gass-Seidel method This method is onl a refinement of Gass-Jacobi method.in this method once a new vale for a nknown is fond it is sed immediatel for compting the new vales of the nknowns. If the r th iterates are then the iteration scheme for this method is r r r r r r r r r b a d c z z c a d b z c b d a Hence finding the vales of the nknowns we se the latest available vales on the R.H.S The process of iteration is contined ntil the convergence is obtained with desired accrac.

18 Conditions for convergence Gass-seidel method will converge if in each eqation of the given sstem the absolte vales o the largest coefficient is greater than the absolte vales of all the remaining coefficients n aii aij i... n j j This is the sfficient condition for convergence of both Gass-Jacobi and Gass-seidel iteration methods. Rate of convergence The rate of convergence of gass-seidel method is roghl two times that of Gass-Jacobi method. Frther the convergences in Gass-Seidel method is ver fast in gass-jacobi.since the crrent vales of the nknowns are sed immediatel in each stage of iteration for getting the vales of the nknowns. Problems.Solve b Gass-Jacobi method the following sstem 8 z z 7 z 5 Rearranging the given sstem as 8 z 7 z z 5 The coefficient matri is is diagonall dominant. Solving for z we have We start with initial vales = The sccessive iteration vales are tablated as follows SCE 9 CIVIL ENGINEERING

19 Iteration X Y Z X=.99;=.57;z=.87.Solve b Gass-Seidel method the following sstem 7 6 z 85;6 5 z 7 5z The given sstem is diagonall dominant. Solving for z we get We start with initial vales = The sccessive iteration vales are tablated as follows Iteration X Y Z X=.5;=.57;z=.96 Eercise:.Solve b Gass-Seidel method the following sstem z 9; z z Ans :.89;.; z.78 SCE CIVIL ENGINEERING

20 .Solve b Gass-Seidel method and Gass-Jacobi method the following sstem z 75; 8 z 7 z 8 Ans : i.;.5; z.5 ii.58;.798; z.69. Matri inversion Introdction A sqare matri whose determinant vale is not zero is called a non-singlar matri. Ever nonsinglar sqare matri has an inverse matri. In this chapter we shall find the inverse of the non-singlar sqare matri A of order three. If X is the inverse of A Then Inversion b Gass-Jordan method B Gass Jordan method the inverse matri X is obtained b the following steps: Step : First consider the agmented matri Step : Redce the matri A in to the identit matri I b emploing row transformations. The row transformations sed in step transform I to A - Finall write the inverse matri A -.so the principle involved for finding A - is as shown below Note The answer can be checked sing the reslt Problems.Find the inverse of b Gass-Jordan method. SCE CIVIL ENGINEERING

21 Ths Hence.Find the inverse of b Gass-Jordan method Eercise:. Find the inverse of b Gass-Jordan method..b Gass-Jordan method find A if. Eigen vale of a Matri Introdction For ever sqare matri A there is a scalar and a non-zero colmn vector X sch that AX X.Then the scalar is called an Eigen vale of A and X the corresponding Eigen vector. We have stdied earlier the comptation of Eigen vales and the Eigen vectors b means of analtical method. In this chapter we will discss an iterative method to determine the largest Eigen vale and the corresponding Eigen vector. Power method Von Mise s power method Power method is sed to determine nmericall largest Eigen vale and the corresponding Eigen vector of a matri A Working Procedre SCE CIVIL ENGINEERING

22 For a sqare matri Assme the initial vector Then find Normalize the vector i.e to get a new vector is the largest component in magnitde of Repeat steps and till convergence is achieved. The convergence of m i and X i will give the dominant Eigen vale Yk i corresponding Eigen vector X Ths lim i... n X And is the reqired Eigen vector. k i and the Problems.Use the power method to find the dominant vale and the corresponding Eigen vector of the matri Let X X X X SCE CIVIL ENGINEERING

23 5 X 5 Hence the dominant Eigen vale is 5.97 and the corresponding Eigen vector is. Determine the dominant Eigen vale and the corresponding Eigen vector of Using power method. Eercise:. Find the dominant Eigen vale and the corresponding Eigen vector of A Find also the other two Eigen vales.. Find the dominant Eigen vale of the corresponding Eigen vector of B power method. Hence find the other Eigen vale also. Ttorial problems Ttorial. Determine the real root of e correct tot for decimal places b Regla falsi method. Ans:.99. Solve the eqation tan=- b Regla falsi method correct to decimal places Ans:.798. Find b Newton s method the real root of log. correct to decimal places Ans:.76. Find the doble root of the eqation b Newton s method. Ans:= 5. Solve tan b iteration method starting with Ans:. Ttorial.Solve the following sstem b Gass-elimination method SCE CIVIL ENGINEERING

24 5 ; 7 ; Ans:.Solve the sstem of eqation z 8 z z b Gass-Jordan method Ans:==z=..Solve the following eqations z 9 z z B Gass-Seidel method..solve b Gass Jacobi method the following 8 z z z 5 Ans:=.99=.7z= Solve the following eqations B Gass-Seidel method. 6 5; 6 Ans: Ttorial.Find the inverse of sing Gass-Jordan method. Ans:.B Gass-Jordan method find A - if Ans:.Determine the dominant Eigen vale and the corresponding Eigen vector of sing power method. Ans:[.6.6] T.Find the smallest Eigen vale and the corresponding Eigen vector of sing power method. Ans:[.77.77] T 5.Find the dominant Eigen vale and the corresponding Eigen vector of method.hence find the other Eigen vale also. Ans:λ=.8 b power SCE 5 CIVIL ENGINEERING

25 QUESTION BANK PART A. What is the order of convergence of Newton-Raphson methods if the mltiplicit of the root is one. Order of convergence of N.R method is. Derive Newton s algorithm for finding the p th root of a nmber N. If = N /p Then p -N = is the eqation to be solved. Let f = p -N f = p p- B N.R rle if r is the r th iterate X r+ = r - = r - = =. What is the rate of convergence in N.R method? The rate of convergence in N.R method is of order. Define rond off error. The rond off error is the qantit R which mst be added to the finite representation of a compted nmber in order to make it the tre representation of that nmber. 5. State the principle sed in Gass-Jordan method. Coefficient matri is transformed into diagonal matri. 6. Compare Gassian elimination method and Gass- Jordan method. Gassian elimination method Coefficient matri is transformed into pper trianglar matri Direct method We obtain the soltion b back sbstittion method Gass- Jordan method Coefficient matri is transformed into diagonal matri Direct method No need of back sbstittion method SCE 6 CIVIL ENGINEERING

26 7. Determine the largest eigen vale and the corresponding eigen vale vector of the matri correct to two decimal places sing power method. AX = = = = X AX = = = = X This shows that the largest eigen vale = The corresponding eigen vale = 8. Write the Descartes rle of signs An eqation f = cannot have more nmber of positive roots than there are changes of sign in the terms of the polnomial f. An eqation f = cannot have more nmber of positive roots than there are changes of sign in the terms of the polnomial f. 9. Write a sfficient condition for Gass seidel method to converge.or State a sfficient condition for Gass Jacobi method to converge. The process of iteration b Gass seidel method will converge if in each eqation of the sstem the absolte vale of the largest coefficient is greater than the sm of the absolte vales of the remaining coefficients.. State the order of convergence and convergence condition for NR method? The order of convergence is Condition of convergence is. Compare Gass Seidel and Gass elimination method?... Gass Jacobi method Convergence method is slow Direct method Condition for convergence is the coefficient matri diagonall dominant Gass seidel method The rate of convergence of Gass Seidel method is roghl twice that of Gass Jacobi. Indirect method Condition for convergence is the coefficient matri diagonall dominant Is the iteration method a self correcting method alwas? In general iteration is a self correcting method since the rond off error is smaller. SCE 7 CIVIL ENGINEERING

27 If g is continos in [a b] then nder what condition the iterative method = g has a niqe soltion in [a b]. Let = r be a root of = g.let I = [a b] be the given interval combining the point = r. if g for all in I the seqence of approimation... n will converge to the root r provided that the initial approimation is chosen in r. When wold we not se N-R method. If is the eact root and is its approimate vale of the eqation f =.we know that = If is small the error will be large and the comptation of the root b this method will be a slow process or ma even be impossible. Hence the method shold not be sed in cases where the graph of the fnction when it crosses the ais is nearl horizontal. 5 Write the iterative formla of NR method. X n+ = n Part B.Using Gass Jordan method find the inverse of the matri.appl Gass-seidel method to solve the eqations +-z=7: +-z=-8: -+z=5..find a positive root of - log X =6 sing fied point iteration method..determine the largest eigen vale and the corresponding eigen vector of the matri with T as the initial vector b power method. 5.Find the smallest positive root of the eqation = sin correct to decimal places sing Newton-Raphson method. 6.Find all the eigen vale and eigen vectors of the matri sing Jacobi method. 7.Solve b Gass-seidel iterative procedre the sstem 8--z=: 6++z=5: +z=. SCE 8 CIVIL ENGINEERING

28 8.Find the largest eigen vale of b sing Power method. 9.Find a real root of the eqation + -= b iteration method..using Newton s method find the real root of log X=. correct to five decimal places..appl Gass elimination method to find the soltion of the following sstem : +-z=5: +-z=: -+z=..find an iterative formla to find where N is a positive nmber and hence find.solve the following sstem of eqations b Gass-Jacobi method: 7+6-z=85:++5z=: 6+5+z=5..Find the Newton s iterative formla to calclate the reciprocal of N and hence find the vale of. 5.Appl Gass-Jordan method to find the soltion of the following sstem: ++z=: ++z=: ++5z=7. SCE 9 CIVIL ENGINEERING

29 . Interpolation with Uneqal intervals Introdction CHAPTER INTERPOLATION AND APPROXIMATION Interpolation is a process of estimating the vale of a fnction at ana intermediate point when its vale are known onl at certain specified points.it is based on the following assmptions: i ii Given eqation is a polnomial or it can be represented b a polnomial with a good degree of approimation. Fnction shold var in sch a wa that either it its increasing or decreasing in the given range withot sdden jmps or falls of fnctional vales in the given interval. We shall discss the concept of interpolation from a set of tablated vales of when the vales of are given intervals or at neqal intervals.first we consider interpolation with neqal intervals. Lagrange s Interpolation formla This is called the Lagrange s formla for Interpolation. Problems. Using Lagrange s interpolation formla find the vale of corresponding to = from the following data X Y 6 Given SCE CIVIL ENGINEERING

30 B Lagrange s interpolation formla Using = and the given data =.67. Using Lagrange s interpolation formla find the vale of corresponding to =6 from the following data X 7 9 Y Given B Lagrange s interpolation formla Using =6 and the given data 6=7.Appl Lagrange s formla to find f5given that f=f=f=8 and f7=8 Given B Lagrange s interpolation formla SCE CIVIL ENGINEERING

31 Using =5 and the given data 5=.9 Eercise.Find the polnomial degree fitting the following data X - Y - - Ans:.Given Ans: Inverse Interpolation b Lagrange s interpolating polnomial Lagrange s interpolation formla can be sed find a vale of corresponding to a given which is not in the table. The process of finding sch of is called inverse interpolation. If is the dependent variable and is the independent variable we can write a formla for as a fnction of. The Lagrange s interpolation formla for inverse interpolation is Problems.Appl Lagrange s formla inversel to obtain the root of the eqation Given that Given that SCE CIVIL ENGINEERING

32 To find sch that Appling Lagrange s interpolation formla inversel we get =.85.Given data Find the vale of corresponding to =. Given that Using the given data and = we get X=8.656 The vale of corresponding to = is SCE CIVIL ENGINEERING

33 . Divided Difference Newton Divided Difference Interpolation Formla Introdction If the vales of are given at neqal intervals it is convenient to introdce the idea of divided differences. The divided difference are the differences of =f defined taking into consideration the changes in the vales of the argment.using divided differences of the fnction =fwe establish Newton s divided difference interpolation formla which is sed for interpolation which the vales of are at neqal intervals and also for fitting an approimate crve for the given data. Divided difference Let the fnction assme the vales Corresponding to the argments... n respectivel where the intervals need to be eqal. Definitions The first divided difference of f for the argments is defined b It is also denoted b [ ] Similarl for argments The second divided differences of f for three argments Is defined as and so on. The third divided difference of f for the for argments Is defined as And so on. SCE CIVIL ENGINEERING

34 Representation b Divided difference table Argments Entr First Divided difference Second D-D Third D-D Properties of divided difference The divided differences are smmetrical in all their argments. The operator is linear. The n th divided differences of a polnomial of n th degree are constants. Problems.If find the divided difference = SCE 5 CIVIL ENGINEERING

35 =.Find the third differences with argments 9 for the fnction = Newton s Divided difference Formla Or Newton s Interpolation Formla for neqal intervals Problems.Find the polnomial eqation passing throgh Given data The divided difference table is given as follows SCE 6 CIVIL ENGINEERING

36 B Newton s formla Is the reqired polnomial.. Given the data 5 f 7 Find the form of the fnction.hence find f. Ans: f=5. Interpolating with a cbic spline-cbic spline Interpolation We consider the problem of interpolation between given data points i i i=.n where a=... n b b means of a smooth polnomial crve. B means of method of least sqares we can fit a polnomial bt it is appropriate. Spline fitting is the new techniqe recentl developed to fit a smooth crve passing he given set of points. Definition of Cbic spline A cbic spline s is defined b the following properties. S i = i i=..n Ss s are continos on [ab] S is a cbic polnomial in each sb-interval i i+ i=..n= Conditions for fitting spline fit The conditions for a cbic spline fit are that we pass a set of cbic throgh the points sing a new cbic in each interval. Frther it is reqired both the slope and the crvatre be the same for the pair of cbic that join at each point. SCE 7 CIVIL ENGINEERING

37 Natral cbic spline A cbic spline s sch that s is linear in the intervals and n i.e. s = and s n = is called a natral cbic spline where s =second derivative at s Second derivative at Note n The three alternative choices sed are n n s and s i.e. the end cbics approach linearit at their etremities. Problems ; n n s s s n s i.e. the end cbics approaches parabolas at their etremities Take s as a linear etrapolation from s and s andsn is a linear etrapolation from s n andsn.with the assmption for a set of data that are fit b a single cbic eqation their cbic splines will all be this same cbic For eqal intervals we have h h h the eqation becomes 6 s i si si [ ] i i i for i=.n- h.fit a natral cbic spline to the following data -8-8 And compte i.5 ii Soltion i Here n= the given data 8 8 i For cbic natral spline s &.The intervals are eqall spaced. s For eqall spaced intervalsthe relation on s s & s is given b 6 s s s [ ] [ h ] s 8 For the interval [ ] the cbic spline is given b a d b c SCE 8 CIVIL ENGINEERING

38 The vales of a b c d are given b a s s 6 s s b c 6 s d [ hi ] a b c d 8 The cbic spline for the interval is s.5 s The first derivative of s is given b h s i si si i 6 h i i Taking i= s s s [ hi ] 6 s We note that the tablated fnction is 9 and hence the actal vales of.5 and are respectivel 5 and. 8.The following vales of and are given obtain the natral cbic spline which agree with at the set of data points 9 Hence compte i.5 and ii Eercise:.Fit the following data b a cbic spline crve Using the end condition that s & s5 are linear etrapolations. SCE 9 CIVIL ENGINEERING

39 .Fit a natral cbic spline to f on the interval [--].Use five eqispaced points of 5 the fnction at =-.hence find.5.. Interpolation with eqal intervals Newton s Forward and Backward Difference formlas Introdction If a fnction =f is not known eplicitl the vale of can be obtained when s set of vales of i i i=.n are known b sing methods based on the principles of finite differences provided the fnction =f is continos. Assme that we have a table of vales i=.n of an fnction the vales of being eqall spaced i e. i ih i... n. i SCE CIVIL ENGINEERING i Forward differences If... n denote a set of vales of then the first forward differences of =f are defined b ; ;.... n n n Where is called the forward difference operator. Backward differences The differences.. n n are called first backward differences and the are denoted b n. 9 ; ;.... n n n Where is called the backward difference operator.in similar wa second third and higher order backward differences are defined. ; ; ; and so on. Fndamental Finite Difference operators Forward Difference operator If =f then f h f Where h is the interval of differencing Backward Difference operator The operator is defined b f f f h Shift operator The shift operator is defined b E r r i.e the effect of E is to shift fnctional vale r to the net vale r.also E r r

40 n In general E r n r The relation between and E is given b E or E Also the relation between and E - is given b E Newton s Forward Interpolation formla Let =f be a fnction which takes the vales... the vales... n where the vales of are eqall spaced. i.e. i ih i... n p p p p p p p.!!... p h This formla is called Newton-Gregor forward interpolation formla. n corresponding to Newton s Backward Interpolation formla This formla is sed for interpolating a vale of for given near the end of a table of vales.let... n be the vales of =f for... n where i ih i... n p p p p p n ph n p n n n...!! n p h This formla is called Newton Backward interpolation formla. We can also se this formla to etrapolate the vales of a short distance ahead of n. SCE CIVIL ENGINEERING

41 Problems.Using Newton s Forward interpolation formla find f. from the following data X..... F Forward difference table: f Let. &. p h p. B Newton s Forward interpolation formla p p. p! =.85. = Using Newton s Forward interpolation formla find the vale of sin5 given that sin 5.77sin sin sin Ans:.788.Using Newton s Backward interpolation formla find when =7from the following data SCE CIVIL ENGINEERING

42 Backward difference table Here n n. n. 9 n. n. n. B Newton s backward interpolation formla n ph n p n p p! n p p p! n... Here =7 p h n p.6 7=.8.Find the cbic polnomial which takes the vale ====.Hence or otherwise obtain.ans:= Eercise:.Given the data 5 78 Find the cbic fnction of sing Newton s backward interpolation formla. Ans:.Using Newton s Gregor backward formla find e.9 from the following data e SCE CIVIL ENGINEERING

43 Ans:.9 e =.96.Estimate ep.85 from the following table sing Newton s Forward interpolation formla e Ans:6.6.Find the polnomial which passes throgh the points 7899sing Newton s interpolation formla. Ttorial Problems Ttorial.Appl Lagrange s formla inversel to obtain the root of the eqation f=given that f=-f=f=9f=5. Ans:.85.Using Lagrange s formla of interpolation find 9.5 given the data X Y 9 Ans:.65.Use Lagrange s formla to find the vale of at =6 from the following data Ans: 7.If log=.77log=.89log5=.8log7=.87find log Ans: Given Find Ans:9 SCE CIVIL ENGINEERING

44 Ttorial.Given the data 5 F 7 Find the form of the fnction. Hence find f.ans:5.find f as a polnomial in powers of-5 given the following table X 7 9 Y Ans: f from the following tablefind f5 X 6 F 88 9 Ans:676.Fit the following data b a cbic spline crve X Y Using the end condition that s and s 5 are linear etrapolations. Ans:-8 5.Given the data X Y Find.5 sing cbic spline fnction. Ans:.89 Ttorial.from the following tablefind the nmber of stdents who obtained less than 5 marks Marks No of stdents 5 5 Ans:8.Find tan.6 from the following valesof tan for.. X Tan Ans:.66.Using newton s interpolation formla find i when =8 ii when =8 from the following data X Y Ans: SCE 5 CIVIL ENGINEERING

45 .Find the vale of f and f from the following data X F Ans:59 5.Estimate sin 8 from the following data given below sin Ans:.7696 Qestion Bank Part A. State the Lagrange s interpolation formla. Let = f be a fnction which takes the vales n corresponding to = n Then Lagrange s interpolation formla is Y = f = o n. What is the assmption we make when Lagrange s formla is sed? Lagrange s interpolation formla can be sed whether the vales of the independent variable are eqall spaced or not whether the difference of become smaller or not.. When Newton s backward interpolation formla is sed. The formla is sed mainl to interpolate the vales of near the end of a set of tablar vales and also for etrapolation the vales of a short distance ahead of. What are the errors in Trapezoidal rle of nmerical integration? The error in the Trapezoidal rle is E< 5. Wh Simpson s one third rle is called a closed formla? SCE 6 CIVIL ENGINEERING

46 Since the end point ordinates and n are inclded in the Simpson s / rle it is called closed formla. 6. What are the advantages of Lagrange s formla over Newton s formla? The forward and backward interpolation formlae of Newton can be sed onl when the vales of theindependent variable are eqall spaced and can also be sed when the differences of the dependent variable become smaller ltimatel. Bt Lagrange s interpolation formla can be sed whether the vales of the independent variable are eqall spaced or not and whether the difference of become smaller or not. 7. When do we appl Lagrange s interpolation? Lagrange s interpolation formla can be sed when the vales of are eqall spaced or not. It is mainl sed when the vales are nevenl spaced. 8. When do we appl Lagrange s interpolation? Lagrange s interpolation formla can be sed when the vales of are eqall spaced or not. It is mainl sed when the vales are nevenl spaced. 9. What are the disadvantages in practice in appling Lagrange s interpolation formla?. It takes time.. It is laborios. When Newton s backward interpolation formla is sed. The formla is sed mainl to interpolate the vales of near the end of a set of tablar vales.. When Newton s forward interpolation formla is sed. The formla is sed mainl to interpolate the vales of near the beginnig of a set of tablar vales.. When do we se Newton s divided differences formla? This is sed when the data are neqall spaced.. Write Forward difference operator. Let = f be a fnction of and let of the vales of. corresponding to of the vales of. Here the independent variable or argment proceeds at eqall spaced intervals and h constantthe difference between two consective vales of is called the interval of differencing. Now the forward difference operator is defined as SCE 7 CIVIL ENGINEERING

47 = =... = These are called first differences..write Backward difference operator. The backward difference operator = For n= = = =. These are called first differences is defined as Part B.Using Newton s divided difference formla find f from the following data and hence find f. 5 f 7..Find the cbic polnomial which takes the following vales: f.the following vales of and are given: f 5 Find the cbic splines and evalate.5 and..find the rate of growth of the poplation in 9 and 97 from the table below. Year X Poplation Y Derive Newton s backward difference formla b sing operator method. 6.Using Lagrange s interpolation formla find a polnomial which passes the points -6. SCE 8 CIVIL ENGINEERING

48 7.Using Newton s divided difference formla determine f from the data: 5 f Obtain the cbic spline approimation for the fnction =f from the following data given that = = The following table gives the vales of densit of satrated water for varios temperatres of satrated steam. Temperatre 5 5 C Densit hg/m Find b interpolation the densit when the temperatre is 75..Use Lagrange s method to find log 656 given that log 65 =.856 log 658 =.88 log 659 =.889 and log 66 =.8..Find f at =.5 and =. from the following data sing Newton s formlae for differentiation Y=f If f=f=f= and f=. Find a cbic spline approimation assming M=M=.Also find f.5..fit a set of cbic splines to a half ellipse described b f= [5- ] /. Choose the three data points n= as and.5 and se the free bondar conditions..find the vale of at = and =8 from the data given below The poplation of a town is as follows: ear poplation thosands Estimate the poplation increase dring the period 96 to976. SCE 9 CIVIL ENGINEERING

49 CHAPTER NUMERICAL DIFFERENTIATION AND INTEGRATION Introdction Nmerical differentiation is the process of compting the vale of For some particlar vale of from the given data i i i=..n where =f is not known eplicitl. The interpolation to be sed depends on the particlar vale of which derivatives are reqired. If the vales of are not eqall spaced we represent the fnction b Newton s divided difference formla and the derivatives are obtained. If the vales of are eqall spaced the derivatives are calclated b sing Newton s Forward or backward interpolation formla. If the derivatives are reqired at a point near the beginning of the table we se Newton s Forward interpolation formla and if the derivatives are reqired at a point near the end of table. We se backward interpolation formla.. Derivatives sing divided differences Principle First fit a polnomial for the given data sing Newton s divided difference interpolation formla and compte the derivatives for a given. Problems.Compte f.5 andf given that f=f=;f= and f5=7 Divided difference table f f f 9 Newton s divided difference formla is SCE 5 CIVIL ENGINEERING

50 f f f f... Here ; ; f f = f ; f f f f 6 f.5.75 f 6..Find the vales f 5 andf 5 sing the following data 7 9 F Ans: 98 and... Derivatives sing Finite differences Newton s Forward difference Formla d d d d d d [ h [ h [ h...]......]...] Newton s Backward difference formla d d n [ h n n n n... ] d d n h [ n n n......] d d n h [ n n......] SCE 5 CIVIL ENGINEERING

51 Problems.Find the first two derivatives of at =5 from the following data Difference table B Newton s Backward difference formla d d n [ h n n n n... ] Here h=; =.5;. n n d d 5.5 d d 5 [ n n h n... =-. SCE 5 CIVIL ENGINEERING

52 .Find first and second derivatives of the fnction at the point = from the following data Difference table We shall se Newton s forward formla to compte the derivatives since =. is at the beginning. For non-tablar vale phwe have d d [ h p p 6 p 6 p 9 p p... Here = h= =. p p. h d d d d SCE 5 CIVIL ENGINEERING

53 Eercise:.Find the vale of sec sing the following data Tan Ans:.78.Find the minimm vale of from the following table sing nmerical differentiation Ans: minimm =.67. Nmerical integration Trapezoidal rle & Simpson s rles Romberg integration Introdction The process of compting the vale of a definite integral from a set of vales i i i=.. Where =a; n b of the fnction =f is called Nmerical integration. Here the fnction is replaced b an interpolation formla involving finite differences and then integrated between the limits a and b the vale is fond. General Qadratre formla for eqidistant ordinates Newton cote s formla On simplification we obtain n d nh n nn n n [ This is the general Qadratre formla B ptting n= Trapezoidal rle is obtained B ptting n= Simpson s / rle is derived B ptting n= Simpson s /8 rle is derived....] Note The error in Trapezoidal rle is of order h and the total error E is given b is the largest of. Simpson s / rle n h d [ n 5... n 6... n ] SCE 5 CIVIL ENGINEERING

54 Error in Simpson s / rle The Trncation error in Simpson s rle is of order h and the total error is given b b a iv iv E h where is the largest of the forth derivatives. 8 Simpson s /8 rle n h d [ 8 n ] Note The Error of this formla is of order h 5 and the dominant term in the error is given b h 5 iv 8 Romberg s integration A simple modification of the Trapezoidal rle can be sed to find a better approimation to the vale of an integral. This is based on the fact that the trncation error of the Trapezoidal rle is nearl proportional to h. Ih I h I h h This vale of I will be a better approimation than I or I.This method is called Richardson s deferred approach to the limit. If h =h and h =h/then we get h I h Problems.Evalate I I sin d b dividing the interval into 8 strips sing i Trapezoidal rle ii Simpson s / rle For 8 strips the vales of =sin are tablated as follows : sin itrapezoidal rle SCE 55 CIVIL ENGINEERING

55 sin d = [ ] 6 =.975 iisimpson s / rle..find d in each case. sin d = [ ] =. b sing Simpsons / and /8 rle. Hence obtain the approimate vale of π We divide the range into si eqal parts each of size h=/6.the vales of point of sbdivision are as follows. /6 /6 /6 /6 5/ B Simpson s / rle d h [ 6 5 [ ] 8 =.785 B Simpson s /8 rle d h [ 6 5 ] 8 Bt d tan [ ] 6 =.785 tan tan ] at each SCE 56 CIVIL ENGINEERING

56 From eqation and we have π=.6 Eercise.Evalate Ans: i. e. Evalate. e d sing i Simpson s / rle ii Simpson s /8 rle taking h=.. d =.8675 ii e d =.867 ii Simpson s / rle Romberg s method Problems cosd b dividing the interval into 8 strips sing i Trapezoidal rle. Use Romberg s method to compte d I correct to decimal places. Hence find log e The vale of I can be fond b sing Trapezoidal rle with= i h=.5 the vales of and are tablated as below: Trapezoidal rle gives I [ ] =.78 i h=.5 the tablated vales of and are as given below: Trapezoidal rle gives I [ = ] ii h=.5 the tablated vales of and are as given below: SCE 57 CIVIL ENGINEERING

57 Trapezoidal rle gives I [ ] =.69 We have approimations for I as I =.78I =.697I =.69 Use Romberg s methodthe better approimation are calclated as follws Using I I I I h I h.69 h h I.69 h h I h.69 B actal integration I d log e log e log e =.69.Use Romberg s method evalate Ans:I=.999 Eercise:.Use Romberg s method to compte approimate vale of Ans:.6. Gassian Qadratre For evalating the integral sin d correct to for decimal places. correct to decimal places.hence find an b I f d we derived some integration rles which reqire the vales of the fnction at eqall spaced points of the interval. Gass derived formla which ses the same nmber of fnction vales bt with the different spacing gives better accrac. Gass s formla is epressed in the form F d w F w F... w F a n n = n i w F i i SCE 58 CIVIL ENGINEERING

58 Where w i and i are called the weights and abscissa respectivel. In this formla the abscissa and weights are smmetrical with respect to the middle of the interval. The one-point Gassian Qadratre formla is given b f d f which is eact for polnomials of degree pto. Two point Gassian formla f d [ f f ] And this is eact for polnomials of degree pto. Three point Gassian formla 8 5 f d f [ f f ] Which is eact for polnomials of degree pto 5 Note: Nmber of terms Vales of t T= & &.5775 Weighting factor Valid pto degree & Error terms iv The error in two point Gassian formla= f and the error in three point Gassian 5 vi formla is= f 575 Note SCE 59 CIVIL ENGINEERING

59 The integral Problems.Evalate eact vale. b a F t dt can be transformed into d B two point Gassian formla t b a b f d b h line transformation a b two point and three point Gassian formla and compare with the f d [ f f ] d =.5 B three-point Gassian formla f d 8 9 f 5 [ 9 f 5 f ] 5 d =.58 d Bt eact vale= [tan ]. 578.Evalate d b sing Gassian three point formla Here f d B three-point Gassian formla 8 5 f d f [ f f ] SCE 6 CIVIL ENGINEERING

60 f f B eqation f d 8 9 f 5 [ 9 f 5 f ] 5 5 = d =.9.Use Gassian two point formla to evalate a=b=. The transformation is b a t b a d t d dt dt = t t B two point Gassian formla f d [ f f ] = =.69 Eercise SCE 6 CIVIL ENGINEERING

61 SCE 6 CIVIL ENGINEERING. Obtain two point and three point Gassian formla for the gass-chebshev Qadratre formla given b d f I Ans: ;. Find d I b Gassian three point formla correct to decimal places. Ans:.5. Evalate cosd I sing Gassian two point and three point formla Ans:.68. Evalate d I sing Gassian here point formla. Ans: 8/. Doble integrals In this chapter we shall discss the evalation of b a d c dd f sing Trapezoidal and Simpson s rle. The formlae for the evalation of a doble integral can be obtained b repeatedl appling the Trapezoidal and simpson s rles. Consider the integral m n dd f I The integration in can be obtained b sccessive application of an nmerical integration formla with respect to different variables. Trapezoidal Rle for doble integral ] [ ] [ f f f f f f f f f hk I Simpson s rle for doble integral The general epression for Simpson s rle for the doble integral can also be derived. In particlar Simpson s rle for the evalation of ] [ ] [ 9 j i j i j i j i j j ij i j i j i j i f f f f f f f f f hk I Problems:

62 . Evalate I Taking h=k=.5 The table vales of e dd sing trapezoidal and Simpson s rle e are given a follows \ i ii Using Trapezoidal rlewe obtain I f dd where f = e hk I [ f [ f f f f f =.76 Using Simpson s rlewe obtain hk [ f f I f f f f 9 =.95 ] f f f ] ] ] f 6 f ].Evalate Taking h=k=.5 dd I b i Trapezoidal rle and ii Simpson s rle with step sizes h=k=.5 The table vales of are given a follows \ SCE 6 CIVIL ENGINEERING

63 SCE 6 CIVIL ENGINEERING i Using Trapezoidal rlewe obtain dd f I where f = ] [ ] [ f f f f f f f f f hk I =.58 i Using Simpson s rlewe obtain ] ] 6 ] [ 9 f f f f f f f f hk I =.57 Eercise. Compte dd I with h=.5 and k=.5 sing Simpson s rle Ans:I=.5. Using Trapezoidal rle evalate dd I taking for sb intervlas Ans:I=.65

64 Ttorial problems Ttorial -.Find the vale of f sing divided differences given data 5 F 8 68 Ans:6.Find the first and second derivatives of the fnction at the point =. from the following data X 5 Y Ans:.7.A rod is rotating in a plane.the angle θ in radians throgh which the rod has trned for varios vales of time t seconds are given below t θ Find the anglar velocit and anglar acceleration of the rod when t=6 seconds. Ans:6.775 radians/sec..from the table given below for what vale of is minimm? Also find this vale of. X Y Ans: Find the first and second derivatives of at =5 from the following dat Ans:.-..Evalate.Evalte Ttorial - d b Simpson s / rle with h=. cos d with h=. b Simpson s / rle..appl Trapezoidal and Simpson s rles to find SCE 65 CIVIL ENGINEERING d taking h=..

65 .Evalate 6 d b dividing the range into eight eqal parts. 5.Using Simpson s rle find log e 7 approimatel from the integral I 7 d.evalate.evalate.evalate. Evalate 5.Evalate d e d d Ttorial - sing Gassian three point formla. sing three-term Gassian Two-point Qadratre formla. sing Gassian three point qadratre formla. cosd sing Gassian two point qadratre formla. sin d sing Gassian two point qadratre formla. Qestion Bank Part A. State the disadvantages of Talor series method. In the differential eqation = f the fnction f ma have a complicated algebraical strctre. Then the evalation of higher order derivatives ma become tedios. This is the demerit of this method.. Which is better Talor s method or R.K method? R.K methods do not reqire prior calclation of higher derivatives of as the Talor method does. Since the differential eqations sing in application are often complicated the calclation of derivatives ma be difficlt. Also R.K formlas involve the comptations of f at varios positions instead of derivatives and this fnction occrs in the given eqation.. What is a predictor- collector method of solving a differential eqation? SCE 66 CIVIL ENGINEERING

66 predictor- collector methods are methods which reqire the vales of at n n- n- for compting the vales of at n+. We first se a formla to find the vales of at n+ and this is known as a predictor formla. The vale of so get is improved or corrected b another formla known as corrector formla.. Define a difference Qotient. A difference qotient is the qotient obtained b dividing the difference between two vales of a fnction b the difference between the two corresponding vales of the independent. d d 5. Write down the epression for & at n b Newton s backward difference formla. d d 6. State the formla for Simpson s /8 th rle. 7. Write Newton s forward derivative formla. 8.State the Romberg s method integration formla fin the vale of 9.Write down the Simpson s /8 rle of integration given n+ data..evalate f d from the table b Simpson s /8 rle. b I f d sing h and h/. a Part B.Appl three point Gassian qadratre formla to evalate..using Trapezoidal rle evalate nmericall with h=. along -direction and k=.5 along -direction..find the first and second derivative of the fnction tablated below at = Using Romberg s method to compte d correct to decimal places.also evalate the same integral sing three point Gass qadratre formla. Comment on the obtained vales b comparing with eact vale of the integral which is eqal to. 5.Evalate sing Simpson s rle b taking h= and k =. 6.Find and at =5 from the following data SCE 67 CIVIL ENGINEERING

67 7.Evalate I= b dividing the range into ten eqal partssing itrapezoidal rle iisimpson s one-third rle. Verif or answer with actal integration. 8. Find the first two derivatives of at =5 and =56 for the given table : Y= / The velocities of a car rnning on a straight road at intervals of mintes are given below: Timemin 6 8 Velocitkm/hr Using Simpson s / rd - rle find the distance covered b the car.. Given the following data find 6 and the maimm vale of if it eists 7 9 Y SCE 68 CIVIL ENGINEERING

68 Chapter Nmerical Soltion of ordinar Differential eqations Talor s series method Introdction An ordinar differential eqation of order n is a relation of the form r n r d F... where = and.the soltion of this differential eqation r d involves n constants and these constants are determined with help of n conditions. Single step methods In these methods we se information abot the crve at one point and we do not iterate the soltion. The method involves more evalation of the fnction. We will discss the Nmerical soltion b Talor series method Eler methods and Rnge - Ktta methods all reqire the information at a single point =. Mlti step methods These methods reqired fewer evalations of the fnctions to estimate the soltion at a point and iteration are performed till sfficient accrac is achieved. Estimation of error is possible and the methods are called Predictor-corrector methods. In this tpe we mainl discss Milne s and Adams-Bashforth method. In the mlti step method to compte n+ we need the fnctional vales n n- n- and n-.. Talor Series method d Consider the first order differential eqation f d The soltion of the above initial vale problem is obtained in two tpes Power series soltion Point wise soltion i Power series soltion +. SCE 69 CIVIL ENGINEERING

69 Where r r d at r d Using eqation the derivatives can be fond b means of sccessive differentiations.epressions gives the vale for ever vale of for which converges. iipoint wise soltion h = +! Where r m r d at r d Where m=. Problems: d. Using Talor series method find at =. if = d -=. Soltion: Given = - and = =h=. Talor series formla for is h = +! = - = =-=- =+ + =+= = + + = ++ + =+ + = + + =+-+ = = = = = +6 + =6-+6+ =-6 SCE 7 CIVIL ENGINEERING

70 = Y.=-.+ + = =.95. Find the Talor series soltion with three terms for the initial vale problem. = +=. Soltion: Given = + = =. = + = + =+= = + + =+ =5 =6+ = + =6+5 = Talor series formla Y= +. = app =+-+ + Eercise:.Find.. given = += [Ans:-.6.5 ] SCE 7 CIVIL ENGINEERING

71 .Find. given = -= [Ans:-.9 ] EULER S METHOD: Introdction In Talor s series method we obtain approimate soltions of the initial vale problem d f as a power series in and the soltion can be sed to compare d nmericall specified vale near. In Eler s methodswe compte the vales of for i ih... with a step h> i.e. where ih... i i Eler s method i = +f n= i Modified Eler s method To compte for i n n hf n n h ii n n [ f n n f n h n error n=.. Error=Oh Note In Eler method n n Where hf Where f slope at f. In modified Eler s method n n Where =h[average of the slopes at and ]Or SCE 7 CIVIL ENGINEERING

72 =h[average of the vales of d at the ends of the interval to ] d.using Eler s method compte in the range.5 if satisfies + =. Soltion: Here f = + = = B Eler s method = +f n= Choosing h=. we compte the vale of sing = +f n= = +hf =+.[+ ] =. = +hf =.+.[.+ ] =.5 = +hf =.5+.[.6+ ] =.675 Eercise: = +hf =.675+.[.9+ ] =.778 = +hf =.778+.[.+ ] =.7 d.given that d Eler method. Ans: and = when =Find when =.5. and.5 sing modified using Eler s methodsolve. Compte. with h=. Ans:.9778 MODIFIED EULER S METHOD:. Given =- =. Using Eler s modified method find.in two steps of. each. Soltion: Let = +h. here and h=. SCE 7 CIVIL ENGINEERING

73 The vale of at = is B modified Eler method = hf f f ] Here f = - f = - = +.= + [f+f.] =+.5[+-.].98 Y.=.98 To find. = =.96 + f f ] =.96-.5[.98+.9] Hence.=.97 Rnge-ktta method The Talor s series method of solving differential eqations nmericall is restricted becase of the evalation of the higher order derivatives.rnge-ktta methods of solving intial vale problems do not reqire the calclations of higher order derivatives and give greater SCE 7 CIVIL ENGINEERING

74 accrac.the Rnge-Ktta formla posses the advantage of reqiring onl the fnction vales at some selected points.these methods agree with Talor series soltions pto the term in h r where r is called the order of that method. Forth-order Rnge-Ktta method d d f This onl is commonl sed for solving initial vales problem Working rle The vale of where h where h is the step-size is obtained as follows.we calclate sccessivel. k k k k hf m hf hf hf m m m m h h h m m m k k k Finall compte the increment m m [ k k k k] where m= 6 Problems d. Obtain the vale of at =. if satisfies d method of forth order. sing Rnge-ktta Here f Let h Choosing h=. =. Then b R-K forth order method SCE 75 CIVIL ENGINEERING

75 SCE 76 CIVIL ENGINEERING ] [ 6 k k k k k h hf k k h hf k k h hf k hf k.=.5 To find where h Taking =. ] [ 6 k k k k k h hf k k h hf k k h hf k hf k.=.7.appl Rnge ktta method to find an approimate vale of for =. in steps of. if d d correct to for decimal places. Here f=+ == Then b R-K forth order method ] [ 6 k k k k

76 k k k k hf hf hf hf h h h. k k k =.65 To find where h Taking =. [ k k k k 6 ] k k k k hf hf hf hf h h h.7 k k k =.76 Eercise: d.use Rnge-ktta method to find when =. in steps of. given that. 5 d Ans: Solve for =...6 b Forht order methodgiven that = when = Ans: SCE 77 CIVIL ENGINEERING

77 Mlti step methodspredictor-corrector Methods Introdction Predictor-Corrector Methods are methods which reqire fnction vales at for the complation of the fnction vale at n+.a predictor is sed to find the n n n n vale of at n+ and then a corrector formla to improve the vale of n+. The following two methods are discssed in this chapter. Milne s method Adam s method. Milne s Predictor-Corrector method Milne s predictor formla is 5 h v And the error = 5 Where n n Milne s corrector formla is And the error = 5 h v 9 Where n n SCE 78 CIVIL ENGINEERING

78 Problems.Using Milne s method compte.8 given that d d.68 We have the following table of vales X Y = To find.8.8 here h=. 8 Milne s predictor formla is where. =.8 =.9 Milne s corrector formla is where. =.9.8=.9. Compte. b Milne s method given that with h=. se Talor s method to find the starting vales. SCE 79 CIVIL ENGINEERING

79 Ans: Adams-bashforth Predictor and Adams-bashforth corrector formla. Adams-bash forth predictor formla is Adams bash forth corrector formla is where. 5 5 iv 9 5 iv The error in these formlas are h f and h f 7 7 Problems.Given Adam s-bashforth method. estimate.8 b To find for. 8here h=. B Predictor formla =.6.76 SCE 8 CIVIL ENGINEERING

80 =.5.8=.5.Given Estimate.8 sing Adam s method To find for. 8here h=. B Predictor formla =.5.75 =.97.8=.97 SCE 8 CIVIL ENGINEERING

81 Ttorial problems Ttorial-.Using Talor\s series method find. approimatel given that correct to for decimal places. d d e = d.compte for =.. correct to for decimal places given d sing talor s series method..using Eler s Modified method find the vales of at =... given d d correct to for decimal places. d. Using Eler s method find.. and.given that d 5.Using Eler s modified method find a soltion of the eqation condition = for the range. 6 in steps of.. d d with the initial Ttorial-.Use R-K forth order method of find at =... if d.appl R-K method of order to solve with h=. d d d.using R-K method to solve to find. and. d d.use the R-K method of forth order find.= given that d d d d d dz 5.Compte and z for =. given z; z d d SCE 8 CIVIL ENGINEERING

82 Ttorial-.Appl Milne s method to find. given that. Use Talor s series method to compte....sing Adam-Bashforth method find the soltion of =.=..=.8.=.997 d d at =.given the vales.using adam s method determine. given that d d.8.given compte... b the forth order R-K method and. b Adam s method. 5.Compte for =... b the forth order R-K method and. b Adam s method if d d SCE 8 CIVIL ENGINEERING

83 Qestion Bank Part A.State Modified Eler algorithm to solve = = at = +h. State the disadvantage of Talor series method. In the differential eqation f = f the fnction f ma have a complicated algebraical strctre. Then the evalation of higher order derivatives ma become tedios. This is the demerit of this method.. Write the merits and demerits of the Talor method of soltion. The method gives a straight forward adaptation of classic to develop the soltion as an infinite series. It is a powerfl single step method if we are able to find the sccessive derivatives easil. If f. involves some complicated algebraic strctres then the calclation of higher derivatives becomes tedios and the method fails.this is the major drawback of this method. However the method will be ver sefl for finding the starting vales for powerfl methods like Rnge - Ktta method Milne s method etc..which is better Talor s method or R. K. Method?or State the special advantage of Rnge-Ktta method over talor series method R.K Methods do not reqire prior calclation of higher derivatives of as the Talor method does. Since the differential eqations sing in applications are often complicated the calclation of derivatives ma be difficlt. Also the R.K formlas involve the comptation of f at varios positions instead of derivatives and this fnction occrs in the given eqation. 5.Compare Rnge-Ktta methods and predictor corrector methods for soltion of initial vale problem. Rnge-Ktta methods.rnge-methods are self startingsince the do not se information from previosl calclated points. SCE 8 CIVIL ENGINEERING

84 .As mesne are self startingan eas change in the step size can be made at an stage..since these methods reqire several evalations of the fnction f the are time consming..in these methodsit is not possible to get an information abot trncation error. Predictor Corrector methods:.these methods reqire information abot prior points and so the are not self starting..in these methods it is not possible to get easil a good estimate of the trncation error. 6. What is a Predictor-collector method of solving a differential eqation? Predictor-collector methods are methods which reqire the vales of at n n- n- for compting the vale of at. n+ We first se a formla to find the vale of at. n+ and this is known as a predictor formla.the vale of so got is improved or corrected b another formla known as corrector formla. 7. State the third order R.K method algorithm to find the nmerical soltion of the first order differential eqation. To solve the differential eqation = f b the third order R.K method we se the following algorithm. and 8.Write Milne s predictor formla and Milne s corrector formla. Milne s predictor formla is Milne s corrector formla is where. where. 9.Write down Adams-bashforth Predictor and Adams-bashforth corrector formla. Adams-bashforth predictor formla is Adams bashforth corrector formla is where..b Talor s series method find. given Ans:.5 SCE 85 CIVIL ENGINEERING

85 Part B.Using Rnge-Ktta method find an approimate vale of for =. if = + given that = when =..Given that + + == = obtain for =.. b Talor s series method and find the soltion for. b Milne s method..obtain b Talor series methodgiven that =+=for =. and. correct to for decimal places..solve for. and z. from the simltaneos differential eqations = +z: =-z; =z=.5 sing Rnge-Ktta method of the forth order. 5.Using Adams method find. given = + =.=..=.58 and.= Using Milne s Predictor-Corrector formla to find. given = =.=.6.=. and.=.. 7.Using Modified Eler s method find. and. if =: =. 8.Given that =+ ;.6=.68.=.8.=.7 = find -. sing Milne s method. 9.Given that =- ; =:.=.8;.=.68 and.6=.779 evalate.8 b Adam s predictor Corrector method..solve b Eler s method the following differential eqation =. correct to for decimal places = with initial condition =. SCE 86 CIVIL ENGINEERING

86 Chapter 5 Bondar vale Problems in ODE&PDE 5. Soltion of Bondar vale problems in ODE Introdction The soltion of a differential eqation of second order of the form F contains two arbitrar constants. These constants are determined b means of two conditions. The conditions on and or their combination are prescribed at two different vales of are called bondar conditions. The differential eqation together with the bondar conditions is called a bondar vale problem. In this chapter we consider the finite difference method of solving linear bondar vale problems of the form. Finite difference approimations to derivatives First derivative approimations h h O h Forward difference h h O h Backward difference h h O h h Central difference Second derivative approimations h h O h h Central difference Third derivative approimations iii [ ] i i i i h Forth derivative approimations SCE 87 CIVIL ENGINEERING

87 iv [ 6 ] i i i i i h Soltion of ordinar differential eqations of Second order i i h i i i i h i Problems. Solve with h=. And h=.5 b sing finite difference method. i Divide the interval[] into two sb-intervals with h=-/=.5 X = =.5 = Let h i i Here.5 B Bondar conditions are We have to determine for =.5 i Replacing the derivative b h i i h [ i i i ] i i SCE 88 CIVIL ENGINEERING

88 Pt i= we get [ ] Using =.5 we get 8.5=.66 ii With h=.5 we have the nmber of intervals = X = =.5 =.5 =.75 = B bondar condition We find the nknown from the relation 6 i i i i i i... Pt i= in 9 Pt i= in 8 7 Pt i= in We have the sstem of eqations Soltion of Gass-Elimination method SCE 89 CIVIL ENGINEERING

89 = The eqivalent sstem is B back sbstittion the soltion is Hence.5=.5.5=.65.75=.85 Eercise d. Solve b finite difference method the bondar vale problem d with = and =. Ans: Solve and with h=.5 5. Soltion of Laplace Eqation and Poisson eqation Partial differential eqations with bondar conditions can be solved in a region b replacing the partial derivative b their finite difference approimations. The finite difference approimations to partial derivatives at a point i i are given below. The -plane is divided into a network of rectangle of lengh h and breadth k b drawing the lines =ih and =jk parallel to and aes.the points of intersection of these lines are called grid points or SCE 9 CIVIL ENGINEERING

90 mesh points or lattics points.the grid points is denoted b ij and is srronded b the i j neighboring grid points i-ji+jij-ij+ etc. Note The most general linear P.D.E of second order can be written as A B C D E F f Where ABCDEF are in general fnctions of and. The eqation is said to be Elliptic if B -AC< Parabolic if B -AC= Hperbolic if B -AC> Soltion of Laplace eqation + = [ i j i j i j i j i j ] This formla is called Standard five point formla [ i j i j i j i j i j This epression is called diagonal five point formla. Leibmann s Iteration Process ] We compte the initial vales of... 9 b sing standard five point formla and diagonal five point formla.first we compte 5 b standard five point formla SFPF. [ b 5 7 b5 b b ] We compte 7. 9 b sing diagonal five point formla DFPF [ b 5 b b5 [ 5 b5 b b7 ] ] SCE 9 CIVIL ENGINEERING

91 [ b 7 5 b5 b [ b 9 7 b b9 5 ] ] Finall we compte 6 8 b sing standard five point formla. [ 5 b [ 5 b5 7 [ b7 [ 8 7 b 9 5 ] ] ] ] The se of Gass-seidel iteration method to solve the sstem of eqations obtained b finite difference method is called Leibmann s method. Problems.Solve the eqation for the following meshwith bondar vaes as shown sing Leibmann s iteration process Let 9 be the vales of at the interior mesh points of the given region.b smmetr abot the lines AB and the line CDwe observe = SCE 9 CIVIL ENGINEERING

92 9 7 9 = 7 = 9 8 Hence it is enogh to find Calclation of rogh vales = =5 =87.5 =7.5 Gass-seidel scheme [5 ] [ 5 ] [ 5 ] 5 [ ] The iteration vales are tablated as follows Iteration No k = 7 = 9 = =. 6 = SCE 9 CIVIL ENGINEERING

93 .When stead state condition prevailthe temperatre distribtion of the plate is represented b Laplace eqation + =.The temperatre along the edges of the sqare plate of side are given b along === along = and =6 along =divide the sqare plate into 6 sqare meshes of side h= compte the temperatre at all of the 9 interior grid points b Leibmann s iteration process. Soltion of Poisson eqation An eqation of the tpe f i.e. is called f poisson s eqation where f is a fnction of and. i j i j i j i j h f ih jh This epression is called the replacement formla.appling this eqation at each internal mesh point we get a sstem of linear eqations in i where i are the vales of at the internal mesh points.solving the eqationsthe vales i are known. Problems.Solve the poisson eqation over the sqare mesh with sides ==== and = on the bondar.assme mesh length h= nit. A B J= J= J= C D I= i= i= i= Here the mesh length h Replacement formla at the mesh point ij i j i j i j i j i j i j SCE 9 CIVIL ENGINEERING

94 Solve the poisson eqation 8 ; and ==== with the sqare meshes each of length h=/. 5. Soltion of One dimensional heat eqation In this chapter we will discss the finite difference soltion of one dimensional heat flow eqation b Eplicit and implicit method Eplicit MethodBender-Schmidt method Consider the one dimensional heat eqation.this eqation is an eample of t parabolic eqation. i j i j i j i j k Where ah Epression is called the eplicit formla and it valid for If λ=/ then is redced into i j [ i j i j ] This formla is called Bender-Schmidt formla. Implicit method Crank-Nicholson method i j i j i j i j i j i j This epression is called Crank-Nicholson s implicit scheme. We note that Crank Nicholson s scheme converges for all vales of λ When λ= i.e. k=ah the simplest form of the formla is given b i j [ i j i j i j i j ] The se of the above simplest scheme is given below. SCE 95 CIVIL ENGINEERING

95 The vale of at A=Average of the vales of at B C D E Note In this scheme the vales of at a time step are obtained b solving a sstem of linear eqations in the nknowns i. Solved Eamples.Solve when t=t= and with initial condition =- pto t=sec assming h t B Bender-Schmidt recrrence relation [ i j i j i j ] ah k For appling eqn we choose Here a=h=.then k= B initial conditions =- we have i i i i B bondar conditions t= == j j SCE 96 CIVIL ENGINEERING