SHIFTED LEGENDRE POLYNOMIAL SOLUTIONS OF NONLINEAR STOCHASTIC ITO - VOLTERRA INTEGRAL EQUATIONS

SHIFTED LEGENDRE POLYNOMIAL SOLUTIONS OF NONLINEAR STOCHASTIC ITO - VOLTERRA INTEGRAL EQUATIONS

S. R. Balachandar, D. Uma, S. G. Venkatesh

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Abstract

In this article, we propose the shifted Legendre polynomial-based solution for solving a stochastic integral equation. The properties of shifted Legendre polyno- mials are discussed. Also, the stochastic operational matrix required for our proposed methodology is derived. This operational matrix is capable of reducing the given sto- chastic integral equation into simultaneous equations with N+1 coe cients, where N is the number of terms in the truncated series of function approximation. These unknowns can be found by using any well-known numerical method. In addition to the capability of the operational matrices, an essential advantage of the proposed technique is that it does not require any integration to compute the constant coe cients. This approach may also be used to solve stochastic di erential equations, both linear and nonlinear, as well as stochastic partial di erential equations. We also prove the convergence of the solution obtained through the proposed method in terms of the expectation of the error function. The upper bound of the error in L2 norm between exact and approximate solutions is also elaborately discussed. The applicability of this methodology is tested with a few numerical examples, and the quality of the solution is validated by comparing it with other methods with the help of tables and  gures.

Keywords

Nonlinear stochastic It^o - Volterra integral equation; shifted Legendre poly- nomial, stochastic operational matrix, convergence analysis; error estimation.