Initial Value Problems for. Ordinary Differential Equations

Transcription

1 Initial Value Problems for Ordinar Differential Equations

2 INTRODUCTION Equations which are composed of an unnown function and its derivatives are called differential equations. It becomes an initial value problem when the original solution is to be found, given the initial condition.

3 Differential equations pla a fundamental role in engineering because man phsical phenomena are best formulated mathematicall in terms of their rate of change.

4 When a function involves one dependent variable, the equation is called an ordinar differential equation or ODE. A partial differential equation or PDE involves two or more independent variables.

5 Differential equations are also classified as to their order. A first order equation includes a first derivative as its highest derivative. A second order equation includes a second derivative, as its highest derivative. Higher order equations can be reduced to a sstem of first order equations, b redefining a variable.

6 Single-step method: calculation at current step onl uses value at last step n n. Multi-step method: calculation at current step uses values at several previous steps.

7 Tpes of Errors : Absolute error : is the numerical difference between the true value and approimate value of a quantit. Truncation errors: The are caused b using approimate results or on replacing an infinite process b a finite one.

8 Round off errors : The arise when a calculated figure is rounded off to a fied number of digits to represent the eact figure, the difference between such rounded figure and eact figure is called round off error.

9 Talor Series Method The method gives a straightforward adaptation of classic calculus to develop the solution of the infinite series. It is a powerful single step method used to find the successive derivative easil. It is useful for finding the starting values for powerful method lie Rungeutta method, Milne's method.

10 Eample : Solve the initial value problem ' -, for at with step length. using Talor series method of order. Solution: Given ' f, -, '' - - 4' ''' -8' - 4' - 4'' iv -' - '' - ''' - 4''' v -48''' - 6''' -'' - 6'''' - 4 iv

11 The fourth order Talor's formula is i h i h ' i, i h '' i, i /! h ''' i, i /! h 4 iv i, i /4!... given at, and with h. we have ' -. '' ''' iv /!. 4 4/4!.965 now at. we have.965

12 ' -.699, '' , '''.997 and iv.995 then /!..997/! /4! /! /! /4!.756

13 /!..696/! /4! /!..955/! /4!.5 now at. we have.5,which is the required solution

14 Euler s Method and Modified Euler s Method The Talor series method ma be difficult to appl if the derivatives become complicated and in this case, it is difficult to determine the error.

15 The error in a Talor series will be small if the step size h is small. In fact, if we mae h small enough, we ma onl need a few terms of Talors series epansion for good accurac. In Euler s method if h is small, the method is too slow and if h is large, it gives in accurate value. It is one of the oldest method.

16 Euler method Formula: i i h f i, i Modified Euler Formula : i i ½ h [f i, i f i h, i hf i, i ] Eample : Find. accurate up to four decimal places using Modified Euler's method b solving the IVP ' -, with step length..

17 Solution: f, - ' -, [.]. with h. Given [.]. Euler Solution: h - [.].

18 Modified Euler iterations:.5 h - - [.]. [.] [.].96 [.].9689 [.].969 Euler Solution: h - [.4].8887

19 Modified Euler iterations:.5 h - - [.4].8887 [.4].866 [.4].8669 [.4] [.4] Euler Solution: h - [.6].74589

20 Modified Euler iterations:.5 h - - [.6] [.6].79 [.6].74 [.6].74 [.6].744 Euler Solution: 4 h - 4 [.8].6866

21 Modified Euler iterations: 4.5 h [.8].6866 [.8].65 [.8].685 [.8].64 [.8].645 Euler Solution: 5 4 h - 5 [.].494

22 Modified Euler iterations: h [.].494 [.].554 [.].57 [.].58 [.].59 [.].5

23 Runge -Kutta method Runge -Kutta methods generate an accurate solution without the need to calculate high order derivatives. Second order Runge-Kutta method have local truncation error of order Oh and global truncation error of order Oh. Higher order Runge-Kutta method have better local and global truncation errors.

24 4 th Order Runge-Kutta Formula: error is global and error is Local 6,,,, i 4 i h O h O h h h f h h f h h f f i i i i i i i i

25 Eample : d,. 5 d Use Runge Kutta method to find.,. 4 Solution: Ste p : h., f,,,. 5

26 h... h h, f... h h, f... h h, f., f

27 Step :.89.,,, f h., h h, f.9 h h, f.98 h h, f, f

28 Summar of the solution : i i

29 Predictor-Corrector Method In numerical analsis, a predictor corrector method is an algorithm that proceeds in two steps. First, the prediction step calculates a rough approimation of the desired quantit. Second, the corrector step refines the initial approimation using another means. There are two such methods namel Milne s method and Adam s method

30 Milne s Method : It is also a multi-step method where we assume that the solution to the given IVP is nown at the past four equall spaced points t, t, t and t. 4h Pr edictor : n n n n n h Corrector : n n n 4 n n

31 Eample 4: Tabulate the solution of d t, dt in the interval [,.4] with h., using Milne s Predictor-Corrector method.

32 Solution : To find the solution at first four points t, t, t and t, as Milne s P-C method is not a self starting method, we shall use R-K method of fourth order to get the required solution and then switch over to Milne s P C method.

33 Thus, taing t, t., t., t. we get the corresponding values using R K method of 4 th order; that is,.,.48 and.997 t... t t

34 Using Milne s predictor formula : 4h P: Before using corrector formula, we compute [ ] t predicted value

35 Finall, using Milne s corrector formula, we compute h C: The required solution is: t

36 Adms-Bashforth method Adams- predictor formula : * h n n n n n 4 n, Adams- corrector formula : h n n n n n 4 n

37 Eample 5: Use the Adams-Bashforth- Method with h. to obtain.8 for the solution of, Solution: With h.,.8 will be approimated b 4. To get started, we use the RK4 method with,, h. to obtain.4,.98796,.646

38 Now with,.,.4, 4.6, and f,, we find Then predictor formula gives , f..., f..., f, f * 4

39 To use the corrector formula, we need * 4 f 4,

40 Applications Runge - Kutta method is applied in man chemical and biological settings. In hdrolog, equations of this tpe model the transport and fate of adsorbing contaminants and microbe-nutrient sstems in groundwater. In chemistr, this equation can stimulate the air pollution in environmental cases.

41 Adams Bashforth scheme used in modeling the population dnamics in erbium-doped fiber amplifiers and lasers based on highl doped fibers. Euler method is used to find the salt concentration in chemical Engineering and to find the voltage across the capacitor in electronic Engineering.