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Runge Kutta Method

The document summarizes numerical methods for solving ordinary differential equations, including: 1) Taylor's series method, which expresses the solution as a Taylor series expansion around an initial point. 2) Picard's method of successive approximation, which iteratively finds better approximations to the solution through integration. 3) Euler's method, which approximates the solution at discrete points using the slope of the tangent line at each point. 4) Modified Euler's method, which improves upon Euler's method by using the average of the slopes over each interval. 5) Runge-Kutta methods, which use multiple slope estimates within each interval to provide greater accuracy than Euler's method.

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350 views7 pages

Runge Kutta Method

The document summarizes numerical methods for solving ordinary differential equations, including: 1) Taylor's series method, which expresses the solution as a Taylor series expansion around an initial point. 2) Picard's method of successive approximation, which iteratively finds better approximations to the solution through integration. 3) Euler's method, which approximates the solution at discrete points using the slope of the tangent line at each point. 4) Modified Euler's method, which improves upon Euler's method by using the average of the slopes over each interval. 5) Runge-Kutta methods, which use multiple slope estimates within each interval to provide greater accuracy than Euler's method.

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Ahtisham ul haq
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Numerical Solution of Ordinary

Differential Equations

Runge-Kutta Method

Numerical Computation

Class: BCS-7 (A, B, C, D)


Instructor: Dr. Tayyab Mahmood
Recall
To describe various numerical methods for the solution of ordinary differential
equations, we consider the general first order differential equation
dy
 f ( x, y ),
dx
with initial condition

y ( x0 )  y0

and illustrate the theory with respect to this equation. The methods so developed can, in
general, be applied to the solution of systems of first-order equations, and will yield the
solution in one of the two forms:
i. A series for y in terms of powers of x, from which the value of y can be obtained by
direct substitution.
ii. A set of tabulated values of x and y.
The methods of Taylor and Picard belong to class (i), whereas those of Euler, Runge—Kutta,
etc., belong to class (ii)
(i) Taylor’s Series Method
We consider the differential equation
dy
 f ( x, y ), y ( x0 )  y0
dx
If y(x) is the exact solution of above equation, then the Taylor's series for y(x) around x  x0
is given by

( x  x0 ) 2 '' ( x  x0 )3 '''
y ( x)  y ( x0 )  ( x  x0 ) y ' ( x0 )  y ( x0 )  y ( x0 )
2! 3!
( x  x0 ) 4 ( iv )
 y ( x0 )  ...
4!
(ii) Picard’s Method of Successive Approximation
We consider the differential equation
dy
 f ( x, y ), y ( x0 )  y0
dx
By integrating the above differential equation, we have found successive approximation
formula for n=0, 1, 2, 3…….
x
y ( n )  y0   f ( x, y ( n 1) )dx with y ( x0 )  y0 .
x0
(iii) Euler Method
We consider the differential equation
dy
 f (t , y ), y (t0 )  y0
dt
Thus. we obtain in general, the solution of the given differential in the form of recurrence
relation
𝑦 = 𝑦 + ℎ𝑓(𝑡 , 𝑦 )

(iv) Modified Euler Method


Instead of approximating 𝑓(𝑡, 𝑦) by 𝑓(𝑡 , 𝑦 ), we now approximate the integral by means of
trapezoidal rule to obtain
( )
𝑓(𝑡 , 𝑦 ) + 𝑓 𝑡 , 𝑦
𝑦 =𝑦 +ℎ
2

We thus obtain the iteration formula


( )
𝑓(𝑡 , 𝑦 ) + 𝑓 𝑡 ,𝑦
𝑦 =𝑦 +ℎ
2

This is modified Euler’s method.


Runge-Kutta Method

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