A NEW APPROACH TO FIND AN APPROXIMATE SOLUTION OF LINEAR INITIAL VALUE PROBLEMS WITH HIGH DEGREE OF ACCURACY

A NEW APPROACH TO FIND AN APPROXIMATE SOLUTION OF LINEAR INITIAL VALUE PROBLEMS WITH HIGH DEGREE OF ACCURACY

U. P. Singh

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Abstract

This work investigates a new approach to  nd closed form solution to linear initial value problems (IVP). Classical Bernoulli polynomials have been used to derive a  nite set of orthonormal polynomials and a  nite operational matrix to simplify deriva- tives in IVP. These orthonormal polynomials together with the operational matrix of relevant order provides a robust approximation to the solution of a linear initial value problem by converting the IVP into a set of algebraic equations. Depending upon the na- ture of a problem, a polynomial of degree n or numerical approximation can be obtained. The technique has been demonstrated through four examples. In each example, obtained solution has been compared with available exact or numerical solution. High degree of accuracy has been observed in numerical values of solutions for considered problems.

Keywords

approximate solution, Bernoulli polynomials, initial value problems, orthonor- mal polynomials.