Solve COMPLETELY for all what is asked. This is Taylor-Series method and Runge-Kutta Fourth Order Method and I have given all necessary methodologies for the solution. This is one of the most difficult problem in Numericals of Applied Mathematics. PLEASE SOLVE COMPLETELY.
Taylor Series Method In vector format, we write the Taylor series method (4.20) of order p as yil = yithy + , yi " + ..+ p! (4.57) where E In component form, we obtain (4.58) (2in = (2+h(y1)+ ,(yz" ),t ...the (4.59) Euler's method for solving the system is given by (m= (y,they), =(1,thfix, (1 (12). (4.60) (4.61) Runge-Kutta Fourth Order Method In vector format, we write the Runge-Kutta fourth order method (4.51) as yil = y, + (k, + 2k, + 2k, + kj), i = 0, 1, 2, ... (4.62) where (4.63) k = hf (x. (je (12;), n=1,2.
Predictor Methods (Adams-Bashforth Methods) All predictor methods are explicit methods. We havek data values, (x, /)(x, 1 / ), ....(x, ,pf ..). For this data, we fit the Newton's backward difference interpolating polynomial of degree & - 1 as (see equation (2.47) in chapter 2) P (x) = flux; + sh) = fix.) + sV/(x)+ s(s + 1) 2! V2 / (x. ) + ... + s(s + 1)(s + 2)... (s + k -2) (k - 1)! v*-I f(x, ). (4.70) Note that s = [(x -x,)h] < 0. The expression for the error is given by T.E. _ S(8 + 1)(s+2)...(s + k- D) (k)! hope (g) (4.71) where [ lies in some interval containing the points r,, x, pax, , and x. We replace /x, y) by P. (x) in (4.69). The limits of integration in (4.69) become for x = x, s = 0 and for x = x, . $ = 1. Also, de = hds. We get
REVIEW QUESTIONS 1. Write the Heun's method for solving the first order initial value problems in the Runge- Kutta formulation. 2. Write the modified Euler method for solving the first order initial value problems in the Runge-Kutta formulation. 3. Why is the classical Runge-Kutta method of fourth order, the most commonly used method for solving the first order initial value problems ?
Solve the initial value problem u'= -3u + 20, w(0) = 0 D = 3u - 40, D(0) = 0.5, with h = 0.2 on the interval [0, 0.4), using the Runge Kutta fourth order method.
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