SKP Engineering College

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1 S.K.P. Engineering College, Tiruvannamalai SKP Engineering College Tiruvannamalai 666 A Course Material on By S.Danasekar Assistant Professor Department of Matematics

2 S.K.P. Engineering College, Tiruvannamalai Quality Certificate Tis is to Certify tat te Electronic Study Material Subject Code:MA659 Subject Name: Year/Sem:II / IV Being prepared by me and it meets te knowledge requirement of te University curriculum. Signature of te Autor Name: S.Danasekar Designation: Assistant Professor Tis is to certify tat te course material being prepared by Mr. S.Danasekar is of te adequate quality. He as referred more tan five books and one among tem is from abroad autor. Signature of HD Name: K. Srinivasan Seal: Signature of te Principal Name: Dr.V.Subramania Barati Seal:

3 S.K.P. Engineering College, Tiruvannamalai MA659 NUMERICAL METHODS OBJECTIVES: Tis course aims at providing te necessary basic concepts of a few numerical metods and give procedures for solving numerically different kinds of problems occurring in engineering and tecnology UNIT I SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS Solution of algebraic and transcendental equations - Fied point iteration metod Newton Rapson metod- Solution of linear system of equations - Gauss elimination metod Pivoting - Gauss Jordan metod Iterative metods of Gauss Jacobi and Gauss Seidel - Matri Inversion by Gauss Jordan metod - Eigen values of a matri by Power metod. UNIT II INTERPOLATION AND APPROXIMATION Interpolation wit unequal intervals - Lagrange's interpolation Newton s divided difference interpolation Cubic Splines - Interpolation wit equal intervals - Newton s forward and backward difference formulae. UNIT III NUMERICAL DIFFERENTIATION AND INTEGRATION Approimation of derivatives using interpolation polynomials - Numerical integration using Trapezoidal, Simpson s / rule Romberg s metod - Two point and tree point Gaussian quadrature formulae Evaluation of double integrals by Trapezoidal and Simpson s / rules. UNIT IV INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS Single Step metods - Taylor s series metod - Euler s metod - Modified Euler s metod - Fourt order Runge-Kutta metod for solving first order equations - Multi step metods - Milne s and AdamsBas fort predictor corrector metods for solving first order equations.

4 S.K.P. Engineering College, Tiruvannamalai UNIT V BOUNDARY VALUE PROBLEMS IN ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS Finite difference metods for solving two-point linear boundary value problems - Finite difference tecniques for te solution of two dimensional Laplace s and Poisson s equations on rectangular domain One dimensional eat flow equation by eplicit and implicit (Crank Nicolson metods One dimensional wave equation by eplicit metod. TOTAL (L:5+T:5: 6 PERIODS OUTCOMES: Te students will ave a clear perception of te power of numerical tecniques, ideas and would be able to demonstrate te applications of tese tecniques to problems drawn from industry, management and oter engineering fields. TEXT BOOKS:. Grewal. B.S., and Grewal. J.S.,"Numerical metods in Engineering and Science", Kanna Publisers, 9t Edition, New Deli, 7.. Gerald. C. F., and Weatley. P. O., "Applied Numerical Analysis", Pearson Education, Asia, 6 t Edition, New Deli, 6. REFERENCES:. Capra. S.C., and Canale.R.P., " for Engineers, Tata McGraw Hill, 5 t Edition, New Deli, 7.. Brian Bradie. "A friendly introduction to Numerical analysis", Pearson Education, Asia, New Deli, 7.. Sankara Rao. K., "Numerical metods for Scientists and Engineers", Prentice Hall of India Private, rd Edition, New Deli, 7.

5 S.K.P. Engineering College, Tiruvannamalai CONTENTS S.No Particulars Page Unit I 6 Unit II Unit III 8 Unit IV 58 5 Unit V 7 5

6 UNIT I SOLUTION OF EQUATIONS AND EIGEN VALUE PROBLEMS Te order of convergence of Newton-Rapson metod is ; Te convergence condition is f (f''( f '( PART A State te order of convergence of Newton Rapson metod. [CO - L - Nov/Dec 5] Write te iterative formula for finding N, by Newton s metod?[co - L - Apr/May 5] N n n n Write down te order of convergence and condition for convergence of fied point iteration metod = g(. [CO - L] Te Order of convergence is One and condition for convergence is g'(, for I were I is te interval containing te root of = g(. State Newton s algoritm for finding square root of. [CO - L- Apr/May 5] N Te Newton s algoritm for is n n, were N= n. =.5, =.6, =.6, =.6 5 If g( is continuous in [a,b], ten under wat condition te iteration metod g( as a unique solution in [a,b]? [CO - L] (i If g( [a,b] for all [a,b], ten g( as a fied point in [a,b]. (ii If ' g ( eists in (a,b and a positive constant k< eists wit g'( k for all (a,b, ten te fied point in [a,b] is unique. 6 State Newton s iterative formula. [CO - L- May/June ] n n f ( n f '( n 7 State fied point teorem.[co - L] Suppose te equation f( = can be written in te form = g(. Let a be a root of f( = and let I be te interval containing a. Suppose g( and ' g ( are continuous in I and g '( a for all, ten te sequence of successive approimations,,, by g( converges to a. 8 Wat is te iterative formula for finding N, were N is a real, by Newton s metod? [CO - H] N n n n i i 6

7 9 Locate te Negative root of =, approimately. [CO - L] Let f( = - + 5; f(- = 6 = positive ; f(- = = positive ; f(- = -6 = Negative; Hence te root lies between - and -. Wat are te merits of Newton s metod of iteration? [CO - H] Newton s metod is successfully used to improve te results obtained by oter metods. It is applicable to te solution of equations involving algebraic functions as well as transcendental functions. Write down te condition for te convergence of Gauss Seidel metod. [CO - L] Te given system of equation a b y cz d, a b y cz d, a b y cz d sould be diagonally dominant. i.e. a b c ; b a c ; c a b For solving a linear system of equations, compare Gauss Elimination metod and GaussJordan metod. [CO - H - Nov/Dec 5] S.No. Gauss-Elimination metod Gauss Jordan metod.. Coefficient matri is transformed into upper triangular matri. We obtain te solutions by back substitution metod. Coefficient matri is transformed into diagonal matri. No need of back substitution metod. Distinguis between direct and iterative metods of solving simultaneous equations.[co - L] S.No Direct Metod Iterative Metod.. We get eact solution. Simple, take less time. Approimate solution. Time consuming, laborious. Self correcting metod. Wat is meant by Diagonally Dominant? [CO - L - May/June ] A matri is diagonally dominant if te numerical value of te leading diagonal element in eac row is greater tan or equal to te sum of te numerical values of te oter element in tat row. i.e. a b c ; b a c ; c a b 7

8 5 By Gauss elimination metod, Solve y, y 5; [CO - L] Te given system equations can be written as AX = B 5 A, B ~ R R R We ave + y =, y =. Hence = y =. y 5 6 Write te procedure to solve AX = B by Gauss Elimination metod. [CO - L] In tis metod te augmented matri is transformed into an equivalent upper triangular matri by elementary row operations and ten by back substitution, we get te solution. 7 Eplain briefly Gauss Jordan iteration to solve te simultaneous equations.[co - L] In tis metod, te augmented matri is reduced to a diagonal matri (or even a unit matri by elementary row operations. Here we get te solutions directly, witout using back substitution metod. 8 Determine te largest eigen value and te corresponding eigen vector of te matri two decimal places using power metod. [CO - L] Let X AX = X AX = X ; AX = X correct to Te largest Eigen value is & te corresponding Eigen vector is. 9 Using Newton s metod, find te root between and of Given f ( 6. 6correct to decimal places. f( = = positive; f( = - = negative Terefore a root lies between and. f ( n f ( 6 s 8

9 Take =; =.6666, =.7, =.7, =.7. 6 Te last two iteration values are equal.te root is.7 Using iteration metod, find te root between and of e correct to two decimal places. Given f ( e f == positive; f( =e- = negative Terefore a root lies between and. e g(, g ' ( e Here g ( < in (, (i Select =.6.Ten = e.677 ; e.687 e Te last two iteration values are equal. Hence =.6 PART B.65 Find Newton s iterative formula to find te reciprocal of a given number N and ence find te Value of. [CO - H] 9 Let N N f N ; i f ( i ; f ( ( i i i i i ( N To find,take =. 5 9 X =.55; =.56; =.56 ; = (ii Find a positive root of te equation 5 = by te metod of fied point iteration. [CO - H]. Te root lies between & = + 5 take =.5 = φ = + 5 =.596, =

10 =.696 =.696 Hence te root is.696 (i Find a positive root of te equation cos by te metod of fied point iteration.[co - H] f( = cos -+= f = +ve f π = ve Te root lies between and π = + cos take =.6 = + cos = =.67 6 =.67 Hence te root is.67 (ii Structure te least positive root of correct to decimal places using Newton Rapson metod. [CO - H- May/June ] Given f(=, f ( f( = -= -ve; f( = = +ve Terefore A root lies between and. Take = f ( n f ( n n ; =.87; f ( n f ( Terefore te root is.856 f ( =.856 ; f ( f ( =.856. f ( (i Find a root of log. Newton s metod correct to tree decimal places. f = log. f = ve, Te root lies between and take = + =. 5 i+ = i f i f i i = =. 765 i = i = =. 765 =. 765 ence te root is.765 f = +ve (ii Find te approimate root of e by Newton- Rapson metod correct to decimal places.

11 [CO - H]. Given f(= e f ( e e f( = -.87 = -ve; f( =.778 = +ve Terefore root lies between and. Here f ( f ( Te root is nearer to Take = f ( n n n ; f ( f ( =.58; f ( n f ( =.99 ; f ( Te value of and are equal. Terefore te root is.99 f ( =.99 f ( (i Solve te following system of equations by Gauss Elimination metod + y z = 5, + y z =, y + z =. [CO - H] 5 (A,B 5 7 R R R R R R 6 8 Hence y z 5 y z 7 6z 8 z y

12 (ii Solve te following system of equations by Gauss-Jacobi metod. [CO - H] 7 6 y z 85, y 5z, 6 5 y z 7; As te coefficient matri is not diagonally dominant we rewrite te equation as 7 6 y z 85, 6 5 y z 7, y 5z ; 85 6 y z 7 y 7 6 z 7 z y 5 Let te initial value be, y, z 5(i ( ( (.8, y.8, z.7 Second iteration: ( ( ( 57, y.69, z.89 ( ( ( Tird iteration:.9,y.685,z Proceeding like ts we get =.6, y=.57, z=.96 Apply Gauss Jordan metod to find te solution of te following system: y z 6, y z 8, y z 9 ; Since te coefficient of in te tird equation is unity, we intercange first and tird equation, we get + y + z = 6, +y +z =8, +y +z = 9. [CO - L - Nov/Dec 5] 6 ( A, B R R R R = R R Terefore =,y=,z=. R R 5(ii Solve y z, y z, y 5z 7 by Gauss Jordan metod.[co - H] + y +5z = 7 +y + z = + y + z =

13 ( A, B R ( R 9 58 R ( R ( 9 R R = 8 8 Terefore =,y=,z= R R R R 8 9 R R 8 8 6(i Solve te following system by Gauss Seidel metod [CO - H- Apr/May 5] y z 7, y z 8, y z 5; 6(ii Te system is diagonally dominant, Now, te system is diagonally dominant. Terefore (7 y z, y ( 8 z, z (5 y Setting y=,z= we get (.85, y (.75, z (.9 ( ( ( Second iteration:.5, y.9998, z ( ( ( Tird iteration:., y., z. Hence =,y= -, z= Test te following system by Gauss Seidel metod y z, 5y z, y 8z ; [CO - H- Apr/May ] Te system is diagonally dominant, Now, te system is diagonally dominant.

14 7(i Terefore ( y z, y ( z, z ( y 5 8 Setting y=,z= we get ( ( (.5, y., z.9 Second iteration: ( ( (.75, y.95, z.965 ( ( ( Tird iteration:.56, y.98, z.995 Proceeding like tis we get =, y=, z=. 6 Using Gauss Jordan metod find te inverse of te matri A Given [ A I ] = A / A / R R /,R R /,R R / / / A 6 / / R R R,R R R 5 / / 6 / / A / / R R 5 / 56 / 56 / 56 / / 6 / 56 A / 56 5 / 56 / 56 R R 6R,R R R 5 / 56 / 56 / 56 Hence A 5 7(ii Using Gauss Jordan metod find te inverse of te matri A

15 Given [ A I ] = A, 6 R R R R R R A / / / R R A, / / / / 6 R R R R R R A, 6 / / / / / 5/ / R R R R R R A Hence 5 6 A A Given [ A I ] = 5 A / R R / 5 / A R R,R R R R / / 7 / / A 5 5 S.K.P Engineering College, Tiruvannamalai 8(i Using Gauss Jordan metod, find te inverse of te matri 5

16 A A A A Hence A / / R R 7 / / / / / / / R 5/ / R 5 7 R R R,R R R R R R, R R R 8(ii A Using Jacobi metod, find te eigen values of a a Solution: cot a 6 5 cot / / 5 5 take sin and cos b a cos a sin a sin 5 5 b a a b ence eigen values are and cos sin R sin cos r 5 Teeigen vectors are X and X 5 6

17 9(i Using Jacobi metod, find te eigen values of cos sin Let R sin cos cot cot a a 6 a 6 A s take sin and cos b a cos a sin a sin 7 6 b a a b ence eigen values of B and A are 7 and. 7 cos sin 6 6 ere B D and Rotation matri R sin cos 6 6 Te eigen vectors are X X 9(ii Find te largest eigen value and eigen vector of, by using Power metod. 7 Let A= AX.Let te initial eigenvector be = X 7 7 ' X 7

18 A ' X AX 7 ' X A ' X AX 7 ' X A X 7 ' X A X 5 ' X A X 6 ' X A X 7 ' X A X 8 ' X A X 9 ' X A X Terefore te eigen vector is.8.56 and te eigen value is 7. (i Find te largest eigen value and eigen vector of 6 A, by using power metod. Also find te Let A= 6.Let te initial eigenvector be X 8 8 S.K.P Engineering College, Tiruvannamalai least latent root and ence find te tird eigen value also. [CO - H- Apr/May 5]

19 6 AX.X 6 7 AX X 6.57 AX X... Proceeding like tis we get, eigen value = and eigen vector = (,.5, To find te least eigen value B = A - I since 6 6 B Let y 6 By. y 6 5 By y 6 5 y y Dominant eigen value is 5 Adding,smallest eigenvalue of A 5 Sum of eigen value Trace of A 6 ( 6 Te eigen values are,, 9

20 5 (ii Find te numerically largest eigen value of A = by using power metod. [CO - H- Apr/May ] and te corresponding eigen vector, ( Let te initial eigenvector be X 5 5 AX ( = ( = 5. 5X AX ( =. = X (.(ie X (..Repeating tis, we.667 get ,5.86.5, Terefore Te largest eigenvalue is 5.8 and te corresponding eigen vector is UNIT II INTERPOLATION AND APPROXIMATION Interpolation is te process of computing intermediate values of a function from a given set of tabular values of te function. PART A List te Interpolation and Etrapolation? [CO - H] Etrapolation is te process of finding te values outside te interval, n. Structure te use of Newton s forward and backward formula? [CO - H- Apr/May 5] Newton s forward formula is used to interpolate value of y nearer to te beginning value of te table. Newton s backward formula is used to interpolate value of y nearer to te end of set of tabular values. Tis may also be used to etrapolate closure to rigt of y n. State any two properties of divided difference. [CO - L- Nov/Dec 5]

21 (i Te divided differences are symmetrical in all teir arguments. i.e. te value of any divided difference is independent of te order of te arguments. (ii Te n t divided differences of a polynomial of degree n are constants. Wat is te relationsip between te Newton s Divided Difference and Newton s forward difference? [CO - H] n n f ( o If te arguments are equally spaced, ten f ( o n n! 5 Construct a linear interpolation polynomial given te points ( o,y o and (,y. [CO - H] ( o f ( yo y o 6 State Lagrange s Interpolation formula for (, y, (, y, (, y.[co - L- Apr/May 5] ( ( ( ( ( ( f ( y y y ( ( ( ( ( ( o o o o o o 7 Write Lagrange s Inverse Interpolation formula. [CO - L] ( y y ( y y...( y yn ( y y ( y y...( y y n o o n o ( y y ( y y...( y yn ( y y ( y y...( y y n o n n o n ( y yo ( y y...( y yn ( y y ( y y...( y y o n... 8 Outline te advantage as lagrange s formula over newton? [CO - H- Nov/Dec 5] Te forwared and backward interpolation formula of newton can be used only wen te value of te independent variable are equally spaced can also be used wen te differences of te dependent variable y become smaller ultimately.but Lagrange s interpolation formula can be used weter te values of, te independent variable are equally spaced or not and weter te difference of y become smaller or not. 9 State te Newton s divided difference formula. [CO - L- Apr/May ] f ( ( f ( o o ( ( o f ( o...(, n ( f ( o, o, State Newton s forward formula. [CO - L] (,... n f ( o,,...

22 u(u u(u (u u(u...(u (n n y( yo u yo yo yo... yo!! n! were u State Newton s Backward formula. [CO - L] v(v v(v (v v(v...(v (n n y( yn vy n yn yn... yn!! n! were v n State te error in Newton s Backward formula. [CO - L- Nov/Dec ] p(p (p...( p n Error f ( p ( n y n n (c were p n (n! Sow tat te second divided difference [,, ] is independent of te order of te arguments. [CO - L] f ( f ( f ( f (,, ( ( ( ( ( ( => f (,, = f (,, = f (,, Tis sows tat f (,, is independent of te order of te arguments. Form te divided difference table for te following data[co - H] : 5 y : f f f Wat is a cubic spline and natural cubic spline? [CO - H] Tird degree polynomials employed to connect eac pair of data points are called cubic splines. If M = and M n =, ten te cubic spline is called as natural cubic spline. 6 State te properties of te cubic spline. [CO - L - Apr/May ] A cubic spline S( is defined by te following properties.

23 (iv S( i = y i, I =,,, n (ii S(, S (, S ( are continuous in [a, b] (iii S( is a cubic polynomial in eac subinterval ( i, i+, I =,,, n 7 Sow tat. bcd a abcd f ( ; f ( a, f ( b, f ( c, f ( d a b c d f ( b, c, d f ( a, b, c f ( a, b, c, d bcd abc bcd a d a d a abcd 8 If f( = f(b f(a (a b f(a,b b a b a b a a b 5 f(, 6 9 Wat are te assumptions we make wen Lagrange s formula is used?[co - H] y, i,,... n i i (ii Te differences of dependent variable are not ultimately small. (iii Te values of, te independent variable need not be in a particular order. (iv All te values of te independent variables must be distinct. f( Prove tat log( f( log f(, find f(a, b and ence f(,. [CO - H] Let y f and, Te assumptions in deriving Lagrange s interpolation formula are: (i Te values of te independent variable need not be equally spaced. log f ( log f ( log f ( log f ( log f ( f ( f ( f ( f ( log f ( f (

24 PART B (i Find te missing term in te following table using Lagrange s interpolation. X y 9-8 ( (8 ( ( ( ( ( ( (9 ( ( ( 5 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( f y y y y f y (ii using Lagrange s formula find te polynomial for te following data ( (7 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( f y y y y f y S.K.P Engineering College, Tiruvannamalai Te Lagrange s interpolation formula f( 7

25 (i Find te tird degree polynomial f( satisfying te following data using Lagrange s formula. Hence find f(.[co - H] 5 7 f( 6 7 ( ( ( ( ( ( y f ( y y ( ( ( ( ( ( ( ( ( ( ( ( y y ( ( ( ( ( ( ere,, 5, 7 & y, y, y 6, y 7 (ii ( ( 5( 7 ( ( 5( 7 ( ( ( 7 ( ( ( 5 y f ( ( ( (6 (7 ( ( 5( 7 ( ( 5( 7 (5 (5 (5 7 (7 (7 (7 5 ( 6 6 f ( Using Lagrange s interpolation calculate te profit in te year from te following data: Year: Profit (in Laks: ( ( y f ( ( ( y 997 y ( 65 y f ( ( ( 999 ( 59 y ( ( ( ( 8 ( ( ( y ( ( ( ( ( ( y y ( ( ( y (i Using Lagrange s interpolation formula, find y( given tat y(5=,y(6=,y(9=andy(=6. [CO - H - Apr/May ] 5

26 ( ( ( ( ( ( y f ( y y ( ( ( ( ( ( ( ( ( ( ( ( y y ( ( ( ( ( ( ( 6( 9( ( 5( 9( y f ( ( ( (5 6(5 9(5 (6 5(6 9(6 ( 5( 6( ( 5( 6( 9 ( (6 (9 5(9 6(9 ( 5( 6( 9 put y(.6666 (ii Using Lagrange s inverse interpolation formula, find te value of wen y= from te given data f, f, f 8, f 8 [CO - H- Apr/May 5] ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y o o o o o o ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y o o y y y y y

27 (i Using Lagrange s inverse interpolation formula, find te value of wen y= from te given data y (ii ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y o o o o o o ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y ( y y o o y y y 8 y 7 y Find f( as a Polynomial in for te following data by Newton s divided differenceformula and ence find f(8. f : 5 7 : f( f( 8 f( f( f(

28 f ( f ( ( f (, ( ( f (,, ( ( ( f (,,, ( ( ( ( f (,,,, f ( ( ( 55 ( ( 5( 7( ence f (8 8 5(i Using Newton s divided difference formula compute f( from te data y f( f( 5 f( f( f( f ( f ( ( f (, ( ( f (,, ( ( ( f (,,, ( ( ( ( f (,,,, 8

29 ere,,,, 5 f( 5 f (, f (,, 9 f (,,, f (,,,, f ( f ( 9 5(ii Using Newton s divided difference, compute f(5 from te data [CO - L - May/June ] 7 y y Δ Δ Δ Δ f ( f ( ( f (, ( ( f (,, ( ( ( f (,,, ( ( ( ( f (,,,, To find f(5: f(5 =. 9

30 6(i Given te table. X: Y: Evaluate f (9 using Newton s divided difference formula. [CO - H- Apr/May ] y Δ Δ Δ Δ f ( f ( ( f (, ( ( f (,, ( ( ( f (,,, ( ( ( ( f (,,,, To find f(9: f(9 = = 8 6(ii Find te missing values from te following table [CO - H] X y 6-7 -

31 y Δ Δ Δ Δ 6 5 a- a- -a a 7-a -+6a+b 7-a -6+a+b b+a -b-a b-7 8-b-a b 8-b -b 5 Since only four values are given, we ave fourt difference is zero Terefore Δ y = 6a+b=, a+b= Solving tis we get a=.5, b=.5 7(i Using Newton s forward interpolation formula, find te polynomial f satisfying te following data and ence find te value of y for = 5.[CO - H] X 6 8 Y 8 6

32 y Δ Δ Δ u = = f ( =. u( u y( yo u yo y! f ( o u( u ( u y! o... 7(ii Ceck Newton s backward interpolation to construct an interpolating polynomial of degree for te data: [CO - H] f f f f , ,. Hence find f Using te following table, we ave v v(v v(v (v y( y y y y ; were v!!! ere ;.5 ; v.5 ence y(... f( /.75858

33 y y y y (i. Using Newton s forward interpolation formula, find te cubic polynomial wic takeste following values: f : : y y y y - - 9

34 8(ii u u(u u(u (u y( y y y y!!! ere ; y ; ; u ence y( 7 6 f( f( From te following table find te value at tan 5 5 ' : tan : u u(u u(u (u y( y y y y!!! u(u (u (u u(u (u (u (u 5 y y...! 5! o y y y y y y -.5 ere u 5 o ence y( o o 5 5 5

35 9(i From te given data, find te number of students wose weigt is between 6 and 7. Weigt in lbs : No of students : y Cumulative frequency Below 5 5 Below 6 7 Below 8 7 Below 7 5 y y y y Below 5 59 u u(u u(u (u u(u (u (u y( y y y y y!!!! 7 ere u.5 ence y(7.597 now we take 6 6 u y(6 7 9(ii Terefore, te no.of students wose weigt below 6=7 Te no.of students wose weigt below 7=.597 No. Of students wose weigt is between 6 & 7=.597-7=5.597=5 approimately. Fit a natural cubic spline to te following data. : y: 5 Also compute y (.5 and y ' (. [CO - H- Apr/May 6] 5

36 Here Let M M o Te cubic spline, in is given by i 6 We ave Mi Mi M i [y i yi y i] for i, M M M 6( M M M 8 Tis reduces to, (taking M, M M M ( i M M 8 ( solving ( & ( we get M, M f( y( [(i M i ( i M i] [(i [y i M i]] ( i[y i M i] ( for i f( [ 5] ; ( for i for i f( [ 5] ; (5 Equation(,(5 & (6 give te cubic spline in eac sub interval f( [ 76 8] ; (6 ence f(.5.75 ence f (.667 f ( [ 6 5] ; (i From te given data, y -8-8 Compute f(.5, f(.75,and f ' ( using cubic splines. [CO - H] Here =, = 6

37 i i i i i i i i i i i i t t d k t d k t t tf f t S f f function is te spline k k k k k k t d d ] ( ( [ ( ( ( , 8 7, 8 ] 7( ( 7( [( ( (( ( ( 8 ( ] ( ( ( [( ( ( ( ( (, S d k d k f f S t i put polynomial tis is cubic spline S 5 9 ( (.7 (.75 ( (.5 ( ( ' ' S f S f S S (ii If (, (, ( ( f f f and f, find a cubic spline approimation, assuming M M 7 7 S.K.P Engineering College, Tiruvannamalai Also find f (.5. [CO - H]

38 solution : Here ; n Let M M o 6 We ave Mi Mi M i [y i yi y i] for i, M M M 6[y y y ] M M M 6[y y y ] Tis reduces to, (taking M, M M M 8 ( M M 8 ( solving (& ( we get M f( 5 ; (, M 76 f( y( [(i M i ( i M i] [(i [y i M i]] ( i[y i M i] ( for i for i f( ; (5 for i f( ; (6 Equation(,(5 & (6 give te cubic spline in eac sub interval ence f(.5.5 UNIT-III NUMERICAL DIFFERENTIATION AND INTEGRATION PART A Give te order of error in te Simpson s rule. (b a Error = E M 8 Hence te error in Simpson s one tird rule is of te order 8

39 dy d y State Newton s formula to find te derivatives, d d at using forward differences. [CO - L]. dy (u (u 6u y' f '( yo yo yo... d!! d y 6u 8u y' ' f ''( y (u y y... were u o o o o d dy d y State Newton s formula to find te derivatives, at d n using backward differences. d [CO - L] dy (v (v 6v y' f '( yn yn yn... d!! d y 6v 8v y'' f ''( y (v y y... were v n n n n d dy d y State Newton s formula to find te derivatives, d d [CO - L - May/June ] y' f '( dy d y o yo y... at = using forward differences. d y y'' f ''( y... o yo yo d dy d y 5 State Newton s formula to find te derivatives, at = d n using backward differences. d [CO - L] dy y' f '( y y n n yn... d d y y'' f ''( yn yn yn... d 6 Wen do you use Newton s divided difference interpolation to find te derivatives? [CO - L] o If values of formula.,... o,, n are not equally spaced ten we can use te Newton s divided difference 9

40 7 Define Numerical Integration. [CO - L] b Te process of computing te values of definite integrals y d from a set of ( n + paired values a ( i,y i, i =,,,,n, were o = a, n = b of te function y = f( is called Numerical integration. 8 Wat is te Geometrical interpretation of Trapezoidal rule?[co - H] Te area of te region enclosed by te curve y = f(, te ais, te ordinates = a and = b by using te area of trapezium. 9 Wy Simpson s one tird rule is called a closed formula? [CO - L - Nov/Dec ] Since te end point ordinates y o and y n are included in te Simpson s one tird rule, so it is called closed formula. Wy Trapezoidal rule is so called? [CO - L] Since we approimate te given integral by te sum of n trapezoids, it is called as Trapezoidal rule. Wat approimation is used in deriving Simpson s rule of integration? [CO - H] Simpson s one tird rule approimates te area of two adjacent strips by te area under a quadratic parabola. Define Quadrature [CO - L- Nov/Dec 5] Te process of evaluating a definite integral from a set of tabulated values of function is called as quadrature. If I =.775 and I =.788 find I using Romberg s Metod. [CO - L] In In By Romberg s Metod I =.785 Evaluate d by Gaussian point formula. [CO - H- Apr/May 5] f (d f f I d = =.69 5 State Gaussian point formula. [CO - L]

41 5 8 f (d f f f ( Given te following data, find y e : : (b a Error y' '(, e d by using Simpson s rule wit = / [CO - L] a b; Order of te error 8 Wat are te errors involved in Simpson s rules for te evaluation of a definite integral of te form b a (b a Error 8 e d Wat is te error in Trapezoidal rule and write down its order.[co - H- Nov/Dec ] f (d. [CO - L] iv y (, f (d f f a b; Order of te error 9 Write down two point Gaussian quadrature formula. [CO - L- May/June ] State Simpson s one-tird rule. [CO - L] n f( d [( y y ( y y... yn ( y y... y n n ] PART B (i Find te first two derivatives of at =5 and = 56, for te following table.[co - L] y =

42 Here y = y y y y By Newton 's forward formula dy dy [ y y y...] d d u dy [. (. (...].55 d 5 By Newton 's backward formula d y [ y y...]. d 5 dy dy [ yn yn y n...].75 d d 56 v d y [ y n y n...]. d 56

43 (ii Find te value of sec from te following data tan Te difference table is tan y y y d (tan sec d By Newton 's formula dy dy [ y y y...] d d u dy dy [. (. (....].69 d d u 8 (i d (tan sec. d sec.7 69 A slider in a macine moves along a fied straigt rod. Its distance cm along te rod is given below various values of te time t seconds. Find te velocity of te slider wen t =.second f(

44 y y y y y 5 y 6 y dy (u (u 6u y ' f '( yo yo yo... d!! dy Velocity at..769 d ' (ii Given te following data, find f 6 and te maimum value of f f : 7 9 : Since te arguments are not equally spaced, we will use Newton s divided difference formula By Newton s divided difference, y = f( = f( + (- f(, + (- (- f(,, +..

45 y = f( Φ f( Φ f( Φ f( Φ f( f( = f ( = + + f (6 = 5. Maimum Value: f( is maimum if f ( = => + + =. But te roots are imaginary. Terefore, tere is no etreme value in te range. (i Given te following values of and y, find dy d and d y at =.5 d y : :

46 y y y y y 5 y 6 y y =.9 and y = -.68 (ii Te table given below reveals te velocity v of a body during te time t. Find its Acceleration at t =. t :..... v : Acceleration = dv dt 6

47 y' y' ' f '( f ''( dy d d y d (u (u 6u y y o o yo...!! 6u 8u y (u y o o yo... were u o t v v v v v Use Newton s forward difference formula, dy (u (u 6u y ' f '( yo yo yo... d!! Acceleration wen t =. is.97 (i Evaluate d Case (i: Wen =.5, y using Romberg s metod. Hence obtain an approimate value for.5 y.8.5 7

48 .5 I [.5 (.8].775 ( Using Trapezoidal rule Case (ii: =.5 ten y I.788 Case (iii : = y I.786 Using Romberg s formula, I I I I = ( I I I I = ( From ( and ( d Evaluation:.785 W.K.T d.785 tan.785 tan tan.785 (ii Using Romberg s metod, evaluate d correct to decimal places. Hence find loge 8

49 [CO - L - May/June ] Case (i: Wen =.5 y.5 y I [.5 (.6666].78 ( Using Trapezoidal rule Case (ii: =.5 ten y I [(.5 ( ].697 Case (iii : = y I [(.5 ( ].69 Using Romberg s formula, I I I I = ( I I I I = ( From ( and ( Evaluation: d.69 9

50 W.K.T d.69 log(.69 loge log.69 loge.69 ( log 5(i 5(ii y = e Evaluate e d using Simpson s rule, coosing =.5. [CO - H- Apr/May 5] y By Simpson s rule, I ( y y ( y ( y y Find te value of =.5 y / log from using Simpson s rule wit.5.[co - H] y I ( y y ( y ( y y. By Simpson s rule, Actual Integration: 5

51 log( log. 6(i By dividing te range into equal parts, evaluate sin d by using Trapezoidal rule. Verify your results by actual integration.[co - H- Nov/Dec 5]. Range = π = π, and y =sin y π By Trapezoidal rule, n o y ( d ( yo yn ( y y y... yn I =.98. Actual Integration: sin d cos 6(ii A curve passes troug te points,,.,.,.,.,.,.,,.,. Area y d Volume y d Find Area: y By Simpson s rule, and.,.. Obtain te area bounded by te curve, te X ais and =,=. Also find te volume of solid of revolution got by revolving tis area about te X ais. [CO - H] 5

52 I ( y y 6 ( y y ( y y y5 Area 7.78 sq. units Find Volume: y Volume y d. (. Volume 8. cubic units 7(i Evaluate sin t dt by using Gaussian two point formula. [CO - H] Transform te variable from to t by te transformation t = b a + b + a Here a =, b = π t = π + π Limit : t π = π( + and dt = πd - 5

53 sin t dt sin ( d were f ( sin ( I f f sin t dt (ii Evaluate d I Transform te variable from to t by te transformation = b a t + ere a =, b = = t + = t + b + a by using two point Gaussian Quadrature. [CO - H] and d = dt Limit : t - 5

54 I d dt / t dt t I f f were f ( t t I d 8(i.5 Evaluate e d by using tree point Gaussian Quadrature. [CO - H - May/June ]. Transform te variable from to t by te transformation = b a t + ere a=., b=.5.65t.85 d.65dt Limit : b + a..5 t - 5

55 .5 (.65t.85. e d e (.65 dt (.65 e (.65t.85 ere f ( t e dt (.65t.85 f (.855 f f.6 5 (.65t e dt f f e d (.65( f. 8(ii Evaluate.. d dy y by using Trapezoidal rule, verify your results by actual integration. Divide te range of and y into equal parts... k. f (, y y y/ By Trapezoidal rule, k I= {sum of values of f at te four corners+(sum of te values of f at te remaining nodes on te boundary+(sum of te values of f at te interior nodes} 55

56 .. d dy.6 y Actual Integration: d dy dy d y y d dy y / / log y log.. (log.(log. log (log.(log..6 9(i Evaluate sin Divide te range of and y into equal parts k f (, y sin( y y / y ddy by using Simpson s rule verify your results by actual integration By Simpson s rule, I =.8 Actual Integration: / / / / sin y ddy (sin cos y cos sin y ddy 56

57 9(ii Evaluate ddy y by using Simpson s rule and Trapezoidal rule taking =.5 and k =.5 y f (, y & k.5 / y By Trapezoidal rule k I= {sum of values of f at te four corners+(sum of te values of f at te remaining nodes on te boundary+(sum of te values of f at te interior nodes} I =. By Simpson s rule I =.8 (i Evaluate f (, y y y y y k.5 ddy by Trapezoidel rule wit =k=.5. [CO - H] / y

58 By Trapezoidal rule, k I= {sum of values of f at te four corners+(sum of te values of f at te remaining nodes on te boundary+(sum of te values of f at te interior nodes} I =.5 (ii Evaluate y ddy y by using Trapezoidal rule taking = k =.5 [CO - H - Nov/Dec ] y / By Trapezoidal rule, k I= {sum of values of f at te four corners+(sum of te values of f at te remaining nodes on te boundary+(sum of te values of f at te interior nodes} I =.7 UNIT IV INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS PART A Write down te Taylor series formula for solving first order ODE. [CO - L- May/June ] yn yn yn yn yn...!!! Using Taylor series metod, find te value of y (., from decimal places. [CO - H- Apr/May 5] y y y yy y y yy (y y y yy 6yy y ; y 8; y 8 dy d iv iv y and y( = correct to 58

59 by using Taylor series formula, y =.5 State te Euler s formula.[co - L]. y y f (, y, y y f (, y, were n n n n dy Find y(. from y given tat y( = by Euler s metod.[co - H- Nov/Dec 5] d n n n n y y f(,y y (.(. 5 Write te formula for modified Euler s metod. [CO - L] yn yn f n, yn f ( n, yn Merits: (i Taylor s formula is easily derived for any order according to our interest. (ii Te values of y( for any ( need not be at grid points are easily obtained. Demerits: (i Tis metod suffers from te time consumed in calculating te iger derivatives. 7 Compare Taylor series metod and RungeKutta metod. [CO - L] 8 6 Write te merits and demerits of te Taylor s series metod of solution.[co - L] (i Te use of R-K metod gives quick convergence to te solutions of te differential equations tan Taylor s series metod. (ii Te labour involved in R-K metod is comparatively lesser. (iii In R-K metod, te derivatives of iger order are not required for calculation as in Taylor series metod. Wat are te advantages of R-K metod over Taylor series metod?[co - H] Te Rungekutta metods are designed to give greater accuracy and tey possess te advantage of requiring only te function values at some selected points on te sub interval. 9 Wat is te truncation error in Taylor s series metod?[co - H] n ( n n Error o y ( o( n! Wat is te error of Euler s metod?[co - H] Error at ' ' ( y (, y o(! Wat are te limitations of Euler s metod?[co - H] i Te attainable accuracy is limited by lengt of step. ii Te metod is slow and as limited accuracy. 59

60 Wat is te error in Fourt order Rungekutta metod?[co - H] Error = y( i+ y i+ and te order of error is ( 5. Wat do we mean by saying tat a metod is self-starting? Not self-starting?[co - H] Iteraion metod is self starting, since we can take value wic lies in te given interval [a, b] in wic te root lies. But Milne s metod is not self starting. Since we sould know any four values prier to te value wic we need. Compare Single-step metod Multi-step metods.[co - L- Apr/May 5] S.No Single-step metod Multi-step metod It requires only te numerical value y i in order to compute te net value y i+ It requires not only te numerical value y i but also atleast four of te past values y i -, y i -,, Taylor series, Euler s and R-K metods are single step metods Milne s, Adam s metods are multi step metods 5 Write down te error in Adam s predictor and corrector formulas.[co - L- Nov/Dec 5] Order of error is 5 ; Error in predictor : ( v y ( ; Error in corrector : 9 y 5 ( v ( 6 Write down te error in Milne s predictor and corrector formulas.[co - L] Order of error is 5 ; Error in predictor : 5 ( v y ( ; Error in corrector : 5 ( v y Write down t order RungeKutta formula to solve second order differential equation. We ave y = z and z = f(, y, z. [CO - L] k f ( y, z l g( y, z,, k l k l k f (, y, z l g(, y, z k l k l k f (, y, z l g(, y, z k f ( y k, z l l g( y k, z l,, ( ( ( ( 6 6 y y y z z z y k k k k z l l l l ( 8 Write down te Milne s predictor and corrector formulas.[co - L] Te predictor formula is y y n, P n ( y ' y n ' n y ' n Te corrector formula is y y ( y ' y ' y ' n, C n n n n 6

61 S.K.P Engineering College, Tiruvannamalai 9 State Adam s predictor and corrector formula. [CO - L- Nov/Dec ] Te predictor formula is y y [55y 59y 7 y 9 y ] i i i i i i Te corrector formula is y y [9y 9y 5 y y ] i i i i i i Compare Adam s Basfort metod wit RungeKutta metod. [CO - L] S.No Adam s Basfort Metod Runge-Kutta Metod Multi step metod Single step metod Need four prior values of y i s Need only te last prior value Does not permits canges in te step size Permits canges in te step size PART B (i dy Solve log ( y, d y( by Euler s metod =., find y(. and y(. [CO - H] (ii (i log Given Data is :, y,. and f(,y ( y Euler s Formula is y y f(, y, n,,,, n Put n = we get, Put n = we get, Solve n n y..log.6 y(. n y.6.log..6. y(. dy y, y( d y Given, y,. and f(, y. Euler s Formula is y y f(,y, n,,,, n n by Euler s metod =., find y(. and y(. [CO - H] n n Put n = we get, ( y y(... 9 Put n = we get, (.9 y y( dy Find te value of y at =. from y, d y( by Taylor s series [CO - H-June ] Given, y,. and y y Taylor s series epansion is y n y n yn y n yn!!!... 6

62 S.K.P Engineering College, Tiruvannamalai (ii For n, y y y y y... (!!! y y / y y y // /// ( / y y y // / // y y y y /// / // /// iv 6y 6y y ( iv y ( y 6 Putting te values in (, we get (. (. (. y(. (.( ( ( ( 6... =.95 6 Obtain te approimate value of y at =. &. for te differential equation dy y e, d y( by Taylor s Series metod. Compare te numerical solution obtained wit te eact solution. [CO - H- Nov/Dec 5] Given, y,. and y y e Taylor s series epansion is y y y y n n n n yn...!!! To find y(.: For n, y y y y y... (!!! dy y ' ( y e d y ' Differentiating y' ( y e successively tree times and putting = y =, we get y '' ( y ' e y '' 9 y ''' ( y '' e y ''' y iv iv y ( y ''' e 5 Putting te values in ( 9 5 y( = (.+.5(. +.5( (. = To find y(.: =., y =.5 For n, y y y y y... (!!! y '' ( y ' e y ''.5 y ''' ( y '' e y ''' 6.5 iv y ( y ''' e iv y 55.5 Putting te values in ( y = y(. =.87 By Eact Integration, 6

63 y e e i.e, y( e e y(..87 y(..8 Error in y =.5 Error in y = -. (i dy Solve sin cos y, d y(.5 by Modified Euler s metod =.5, find y(.5 [CO - H] Given Data is :.5, y,.5 and f(,y sin cos y yn yn f n, yn f ( n, y n.5.5 Put n = we get y y.5f (, y [f (, y ] y.5 f (.5.5,.5[ f (.5,] Put n = we get y y ( f, y [ f (, y] y.65.5 f(.5,.65.5[f(,.65].955 (ii (i dy Solve ( y, y( by Modified Euler s metod by coosing =., find y(. and d y(.. [CO - H] y Given = y,. and f(,y. Modified Euler s formula is yn yn f(n,yn [f(n,yn] Put n = we get y y... f (, y [ f (, y].. y y(.. f (, [ f (,].978 Put n = we get.. y y(. y. f (, y [ f (, y].855 Apply Runge Kutta metod, to find an approimate value of y wen =. given tat dy y, y(.[co - L- Apr/May 5] d Given:, y,.and f (, y y 6

64 Finding y y(. : R K metod (for n = is: y y(. y k k k k ( 6 k.. k f (, y. [ ] =. ; k f, y. =. k.. k f, y. =. k f y k... =.888, Using te values of k, k, k and k in (, we get y y(.. ( =.68 6 Hence te required approimate value of y is.68 (ii 5(i Using Runge Kutta metod of fourt order, solve dy y, y ( d y y Given:, y,.and f (, y y Finding y y(. R K metod (for n = is: y y(. y k k k k ( 6 k k f (, y. f (, =. ; k f, y. f.,. =.967 k k f, y. f.,.96=.967; k f y k..,.967 =.89, f Using te values of k, k, k and k in (, we get y y(.. (.967 ( = = Hence te required approimate value of y is.9599 dy Given y, y (, y(. =., y(. =.795 and y(.6 =.76. Compute y( using d Milne s Metod. [CO - L] at =..[CO - H] dy y f (, y d To find y(.8: Given:, y and f f (, y ;., y. and f f (, y.996 6

65 ., y.795 and f f (, y.97 ;.6, y.76 and f f (, y y y( y(.8 To Find : Predictor Metod (P (. y y(.8 y f f f ( =.9 Now we compute f f(.8,.9.77 Corrector Metod ( y C. y(.8 y f f f =.6 Finding y(. Given:., y. and f f (, y.996 ;., y.795 and f f (, y. 97.6, y.76 and f f (, y.56895;.8, y.6 and f f (, y. 77 y 5 y( 5 y(. To Find : Predictor Metod ( (. y P 5 y(. y f f f. ( =.55 Now we compute f 5 f (., Corrector Metod ( y C. 5 y(. y f f f =.556 5(ii Using Milne s metod to find y(. given tat 5y y given tat y(, y(..9, y(..97, y(... [CO - H] y y',,.,.,. 5., y, y.9, y.97, y. Substituting tese values, y.9, y.67, y.5 ' ' ' By Milne s predictor formula ' ' ' y,p y y y y.897 ' y.7 By Milne s corrector formula 65

66 ' ' ' y,c y y y y.87 y y(..87 dy y, y (, y (.., y (..58, y d ( (i Given Evaluate y (. by Adam s Basfort metod.[co - H- Apr/May 5] Given:, y ;., y. ;., y. 58 ;., y. 979 ' y, ' y.79, To Find y y( y(. : ' y.669 and ' y 5.5 Predictor Metod y, P y(. y 55 y 59 y 7 y 9y (..979 (555.5 ( (7.79 (9 =.579 Now we compute y 7. 7, p Corrector Metod y, C y(. y 9 y 9y 5y y..979 (9 7.7 (9 5.5 ( = (ii Using Adam s Basfort metod, find y(. given tat 5y y given tat y(, y(..9, y(..97, y(... [CO - H] y y',,.,.,. 5., y, y.9, y.97, y. Substituting tese values, y.9, y.67, y.5 ' ' ' By Adam s Basfort predictor formula ' ' ' ' y, p y 55y 59y 7y 9y.86 ' y.7 By Adam s Basfort corrector formula 66

67 S.K.P Engineering College, Tiruvannamalai ' ' ' ' y, c y 9y 9y 5y y.87 y y(..87 dy 7 Find te value of y(. by Milne s metod for y y, y(. Use Taylors series to get te d value of y at =., Euler s metod for y at =. and Runge - Kutta t order metod for y at =. [CO - H- Nov/Dec 5] To find y = y(. by Taylor s metod: Given, y,. and y y y Taylor s series epansion is y y y y n n n n yn...!!! For n, y y y y y...!!! ( At (, y (, ' '' ''' y y y y y'' y' y yy' y y''' y'' y' yy'' (y' y Putting te values in ( y = y(. =.67 To find y = y(. by Euler s metod: =., y =.67 Given f(,y y y y Euler s Formula is y Put n = we get, y y f (, y n y n f( n,y n, n,,,, y y(..67 (.((.(.67 ( To find y = y(. by R-K metod:., y.56,.and f (, y y y R K metod (for n = is: y y(. y y were y [k k k k ] ( 6 k f (, y.89 k k f, y. k k f, y

68 k f, y k.67 y.8 y y(. To find y = y(. by Milne s metod: y y y,.,.,.., y, y.67, y.56, y.77 Substituting tese values, y, y.587, y.895, y.6 By Milne s predictor formula ' ' ' y,p y y y y.8 By Pr edictor formula y.8 ' y y y.7 By Milne s corrector formula ' ' ' y,c y y y y.79 y y(..79 ' ' ' ' dy 8 Consider te IVP y, y(.5 d a. Using te modified Euler metod, find y(. b. Using R.K.Metod of order, find y(. and y(.6 Using Adam- Basfort predictor corrector metod, find y(.8 [CO - H- Nov/Dec ] dy Given y d, y.5,., f, y y (a By modified Euler s metod y y f, y f, y (. y f, y y y y.5. f, f., y y

69 Now (b t Runge-kutta metod dy y f, y d., y.88,. K f, y. y k..58 K f, y. f.,.88. f., , k K f y. f.,.88. f., K f, y k. f.., y k k k k y y..88 y , y.,. y?,.., K f y f k.. K f, y. f.,...5, f 69

70 k..7 K f, y. f.,.. f.5, ,..., , K f y k f y y.6 y y y k k k k (iii Using Adam-Bas fort predictor corrector metod y.5 y.88 y. y.677 y? Adam s predictor formula is ' ' ' ' y, n p yn 55y n 59yn 7yn 9yn Put n ' ' ' ' y, p y 55y 59y 7y 9y y y '.5.5 ' y y y y ' ' y y y, p Corrector formula 7

71 y, c y 9y 9y 5y y Put n ' ' ' ' n n n n n n ' ' ' ' y, c y y y y y y y ' Consider te initial value problem dy y d, y( = a. Find y(. and y(. by R.K.Metod of order b. Find y(. by Euler s metod c. Find y(. by Milne s predictor-corrector metod. [CO - H] Given dy y d, y,., f, y y To find y(. = y by R-K metod: R K metod (for n = is: y y(. y k k k k ( 6 k f (, y (.f (, (.(. k k k f, y.9875 k f, y k k f, y.75 Using te values of k, k, k and k in (, we get y y(. (. (.75 ( To find y = y(. by t Runge-kutta metod: 7

72 (i dy y f, y d., y.8,..98,...8. K f y y k K f, y.7 k..7 K f, y. f.,.8.9 K f, y k. f.., y k k k k.77 6 y y. y y y.86 To find y = y(. by Euler s metod: Euler s Formula is yn yn f(n,yn, n,,,, Put n = we get, y y f(,y y y(..86.(.86 (..66 To find y = y(. by Milne s metod: y y,.,.,.., y, y.8, y.86, y.66 Substituting tese values, y, y.98, y.786, y.66 By Milne s predictor formula ' ' ' y,p y y y y.6797 By Pr edictor formula y.6797 ' y y.797 By Milne s corrector formula ' ' ' y,c y y y y. y y(.. Consider te second order IVP Let t= ' ' ' ' y y t y e sin t, wit y ( = -. and y (=-.6. Using Taylor series appr., find y(. and using R.K.Metod of order, find y(.. [CO - H- May/June ] 7

73 S.K.P Engineering College, Tiruvannamalai y y y e sin, y(., y'(.6 ' Also y.6 To find y(. by Taylor s series: ' At (, y (,. & y.6 y e sin y y '' y. y''' e cos e sin y'' y' ''' y. (iv (iv y e (sin cos y''' y'' y 7.6 Taylor s series epansion is y y y y n n n n yn...!!! For n, y y(. y y y y...!!! y(.=.67.6 To find y(. = y by R.K. Metod: ' '' ' Set y z y z z ' z y e sin., y.6, z y ' ' '' y( y y y...!. y.6 z.6 ' dy dz d d k f ( y, z.6 l g( y, z. f (, y, z y ' z; g(, y, z z ' z y e sin,, k l k l k f (, y, z.67 l g(, y, z.7 k l k l k f (, y, z.65 l g(, y, z. k f (, y k, z l.66 l g(, y k, z l.66 y ( k ( k k k.657 z ( l ( l l l. 6 6 y y y.6 ( z z z.6 (..66 y(..67 & y(..557 (ii Solve y.( y y y subject to y(, y ( using fourt order 7

74 RungeKutta Metod. Find y(. and y (. using step size. To find y(. = y and y'(. y by R.K. Metod: ' ' '' ' Set y z y z ' (.( z y z y, y, z y ' dy dz f (, y, z y ' z; g(, y, z z ' (.( y z y d d k f ( y, z (. z.(. l g( y, z.((.( y z y,,.((.( (. k l k l k f (, y, z. l g(, y, z. k l k l k f (, y, z.9999 l g(, y, z. k f (, y k, z l.999 l g(, y k, z l.8 y ( k ( k k k.6 z ( l ( l l l y y y.6.6 z z z ( y(..6 & z y '( l UNIT-V BOUNDARY VALUE PROBLEMS IN ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS Solve y +y=, y( =, y(= wit =.5. [CO5 - H] [ yi yi yi ] i y [ i i yi yi yi ] yi. Wen i =, y = y(.5 =.66 PART A Write te forward, central and backward finite difference for u ui, j ui, j u ui, j ui, j ui, j ui, j u u,, State te central difference approimations for y i i y y y y y y and y i i i i i i i ' and y '' 7

75 State te forward difference approimations for y and y i i y y y y -y y and y i i i i i i i State te backward difference approimations for y and y 5 i i y y i i- y -y y i i i- y and y i i 6 State te general form of Poisson s equation in partial derivatives. u u u u u u u u i, j i, j i, j i, j i, j i, j G(, y or G(, y i j y k 7 State Crank-Nicolson s implicit sceme for one dimensional eat equation. (u u ( u i, j i, j i, j ka ( u (u u were i, j i, j i, j 8 Write te eplicit formula to solve one dimensional wave equation. u ( a u a (u u u i, j i, j i, j i, j i, j ' ' '' '' a a If i.e, k ten u i, j ui, j ui, j ui, j 9 Write down te simplest form of Crank-Nicolson s formula. ui, j ui, j ui, j ui, j If i.e, k ten u i, j a State Bender-Scmidt formula for solving one dimensional eat equation. k ui, j u i, j ( ui, j ui, j were a Write down tesimplest form of Bender-Scmidt formula. ui, j ui, j If i.e, k ten u i, j a Obtain finite difference sceme to solve u + u yy = ui, j ui, j ui, j ui, j ui, j ui, j k 75

76 Write down te classification of linear second order partial differential equation? [CO5 - L] Te linear second order partial differential equation Au + Bu y + Cu yy + Du + Eu y + Fu + G = can be classified as (i Elliptic if B AC < (ii Parabolic if B AC = (iii Hyperbolic if B AC > Classifyu yuy uyy u u Here B AC = y. Hence te given i PDE is Elliptic if y < ii PDE is Parabolic if y = iii PDE is Hyperbolic if y > 5 Write down te central finite difference formula for u yy ui, j ui, j ui, j ui, j k 6 Write down te Standard five point formula ui, j ui, j ui, j ui, j ui, j 7 Write down te Diagonal five point formula u u u u u i, j i, j i, j i, j i, j 8 Write down te Liebmann s iteration process formula: u u u u u (n (n (n (n (n i, j i, j i, j i, j i, j 9 Classifyu uy uyy Here B AC = 6 = PDE is Elliptic. '' Obtain te finite difference sceme for te differential equation y y 5 yi yi yi 5 yi y ( y y 5 i i i 76

77 PART B (i Solve ( y y y given tat y(, y( ( take = =/ =/ = Rewrite te equation as Substitute ( i yi i yi yi. yi+- y i + yi y i = yi yi & yi 9( y - y + y y y y. and simplify we get i i+ i i- i i i i Put i = we get 6y 6y 9( y - y + y y y y ( 6 y.65 y.769 ( 9( y - y + y y y y Put i = we get 7y y.5 y = y =? y =? y =/ (7 y.59 y.6765 ( Solving ( and ( we get y =.55 and y =. (ii Using te finite difference metod, Solve y" + y = subject to y( =, y( = at.5,.5 &.75 y i yi i &.5 yi+- y i + yi y i = yi yi & yi y y ( y Rewrite te equation as Substitute i i i i yi yi yi i 6 6 Put i = we get y.975y y y y.65(.5.975y y.56 ( Put i = we get and simplify we get 77

78 y.975y y.65 y.975y y.65(.5 y.975y y. ( Put i = we get y.975y y.65 y.975y.65(.75 y.975y.95 ( Solving (, ( and ( we get y =.5, y =.7 and y =.56 (i Solve u = u t, given tat u(, = ( -, u(,t =, u(,t = by Bender Scmidt s formula Compute u up to 5 times steps. (taking =, k = [CO5 - L - Nov/Dec 5] Bender Scmidt s formula is u i, j λ(ui, j ui, j ( u i, j (i.e E = (A+C + (- B Here =, =, k = kα = Put = in above formula we get u i, j (ui, j ui, j ui, j A B C D E F (i.e. E = (A+C B t