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GDB Topic:

Dear Students you have studied the following methods to find the numerical solution of the first order ordinary initial value problems:

• Euler’s Method
• Modified Euler’s Method
• Taylor’s Series Method
• Runge- Kutta Method

So, which method should be preferred to improve the accuracy of the numerical solution of first order ordinary initial value problems?  The answer should be Précised (not be exceeded more than two or three lines) and supported with appropriate reasoning(s).

Note: If the answer will not be précised and contains irrelevant material then it will not be entertained.

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### Replies to This Discussion

Shabashhh ha koi Layak Banda

modified euler method is used to omprove the accuracy of numerical solution.

The Euler forward scheme may be very easy to implement but it can't give accurate solutions. A very small step size is required for any meaningful result. In this scheme, since, the starting point of each sub-interval is used to find the slope of the solution curve, the solution would be correct only if the function is linear. So an improvement over this is to take the arithmetic average of the slopes at xi and xi+1(that is, at the end points of each sub-interval). The scheme so obtained is called modified Euler's method. It works first by approximating a value to yi+1 and then improving it by making use of average slope.

I think this will be helpful Runge- Kutta Method is most efficient method to improve the accuracy of numerical solution... Read at page number 202 of MTH603..

in my opinion Runge Kutta Method gives more accuracy as we have to give preference to improve the accuracy.

Thanx brother for sharing your idea. It will be helpful for us

In numerical analysis, the Runge–Kutta methods (German pronunciation: [ˌʁʊŋəˈkʊta]) are an important family of implicit and explicit iterative methods, which are used in temporal discretization for the approximation of solutions of ordinary differential equations.

please also dicuss about this method and give your views Comparison of Euler Modified Euler & Runge Kutta Methods

 Exact Euler Modified Euler Runge-Kutta Order Four 0.0 0.5000000 0.5000000 0.5000000 0.5000000 0.1 0.6574145 0.6554982 0.6573085 0.6574144 0.2 0.8292986 0.8253385 0.8290778 0.8292983 0.3 1.0150706 1.0089334 1.0147254 1.0150701 0.4 1.2140877 1.2056345 1.2136079 1.2140869 0.5 1.4256394 1.4147264 1.4250141 1.4256384

These are computationally, most efficient methods in terms of accuracy. They were
developed by two German mathematicians, Runge and Kutta.
They are distinguished by their orders in the sense that they agree with Taylor’s series
solution up to terms of h
r
, where r is the order of the method. These methods do not
demand prior computation of higher derivatives of y(t) as in TSM.
Fourth-order Runge-Kutta methods are widely used for finding the numerical solutions of
linear or non-linear ordinary differential equations, the development of which is
complicated algebraically.
Therefore, we convey the basic idea of these methods by developing the second-order
Runge-Kutta method which we shall refer hereafter as R-K method.

In mathematics and computational science, the Euler method is a SN-ordernumerical procedure for solving ordinary differential equations (ODEs) with a giveninitial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book Institutionum calculi integralis(published

kia yeee thik h? Comparison of Euler Mehtod and Runge-Kutta method:

https://jmckennonmth212s09.wordpress.com/2009/02/03/runge-kutta-met...

Since we have no hope of solving the vast majority of differential equations in explicit,
analytic form, the design of suitable numerical algorithms for accurately approximating
solutions is essential. The ubiquity of differential equations throughout mathematics and
its applications has driven the tremendous research effort devoted to numerical solution
schemes, some dating back to the beginnings of the calculus. Nowadays, one has the
luxury of choosing from a wide range of excellent software packages that provide reliable
and accurate results for a broad range of systems, at least for solutions over moderately
long time periods. However, all of these packages, and the underlying methods, have their
limitations, and it is essential that one be able to to recognize when the software is working
as advertised, and when it produces spurious results! Here is where the theory, particularly
the classification of equilibria and their stability properties, as well as first integrals and
Lyapunov functions, can play an essential role. Explicit solutions, when known, can also
be used as test cases for tracking the reliability and accuracy of a chosen numerical scheme.
In this section, we survey the most basic numerical methods for solving initial value
problems. For brevity, we shall only consider so-called single step schemes, culminating in
the very popular and versatile fourth order Runge–Kutta Method.

Euler’s Method
The key issues confronting the numerical analyst of ordinary differential equations
already appear in the simplest first order ordinary differential equation. Our goal is to
calculate a decent approxiomation to the (unique) solution to the initial value problem
du
dt
= F(t, u), u(t0) = u0.
To keep matters simple, we will focus our attention on the scalar case; however, all formulas
and results written in a manner that can be readily adapted to first order systems — just
5/18/08 165 c 2008 Peter J. Olver
replace the scalar functions u(t) and F(t, u) by vector-valued functions u and F(t, u)
throughout. (The time t, of course, remains a scalar.)

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