Mathematics IV Notes eBook

Mathematics IV Notes eBook
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Mathematics IV Notes eBook

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obtained rapidly. In this chapter we focus on basic numerical methods for solving initial value problems. Analytical methods, when available, generally enable to find the value of y for all values of x. Numerical methods, on the other hand, lead to the values of y corresponding only to some finite set of values of x. That is the solution is obtained as a table of values,
Unit I Numerical Methods Prepared by Dr. M. Sankar, SCE, Bangalore
rather than as continuous function. Moreover, analytical solution, if it can be found, is exact, whereas a numerical solution inevitably involves an error which should be small but may, if it is not controlled, swamp the true solution. Therefore we must be concerned with two aspects of numerical solutions of ODEs both the method itself and its accuracy. In this chapter some methods for the numerical solution of ODEs are described. The general form of first order differential equation, in implicit form, is dy f x, y . An Initial Value Problem IVP F x, y, y 0 and in the explicit form is dx consists of a differential equation and a condition which the solution much satisfies or several conditions referring to the same value of x if the differential equation is of higher order . In this chapter we shall consider IVPs of the form dy 1 f x, y , y x0 y0 . dx Assuming f to be such that the problem has a unique solution in some interval containing x0, we shall discuss the methods for computing numerical values of the solution. These methods are step-by-step methods. That is, we start from y0 y x0 and proceed stepwise. In the first step, we compute an approximate value y1 of the solution y of 1 at x x1 x0 h. In the second step we compute an approximate value y2 of the solution y at x x2 x0 2h, etc. Here h is fixed number for example 0.1 or 0.001 or 0.5 depends on the requirement of the problem. In each step the computations are done by the same formula. The following methods are used to solve the IVP 1 . 1. 2. 3. 4. 5.
1.
Taylor s Series Method Euler and Modified Euler Method Runge Kutta Method Milne s Method Adams Bashforth Method
Taylor s Series Method
dy