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.

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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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