On dual Bernstein polynomials and stochastic fractional integro‐differential equations

Abstract

In recent years, random functional or stochastic equations have been reported in a large class of problems. In many cases, an exact analytical solution of such equations is not available and, therefore, is of great importance to obtain their numerical approximation. This study presents a numerical technique based on Bernstein operational matrices for a family of stochastic fractional integro-differential equations (SFIDE) by means of the trapezoidal rule. A relevant feature of this method is the conversion of the SFIDE into a linear system of algebraic equations that can be analyzed by numerical methods. An upper error bound, the convergence, and error analysis of the scheme are investigated. Three examples illustrate the accuracy and performance of the technique.

Orthonormal Bernoulli Polynomials for Solving a Class of Two Dimensional Stochastic Volterra–Fredholm Integral Equations

“…In this section, we briefly introduce the Itô integral and its properties. The reader can refer to references [8,16,17] for more information. Definition 1.1 [8,18] For t ∈ [0, T ], B(t) is called the Brownian motion, if it satisfies the following properties:…”

“…then, the Itô integral of g [17] is defined as follows: 4 One of the valuable properties of Itô integral is as follows [17]:…”

“…In this section, we briefly introduce the Itô integral and its properties. The reader can refer to references [ 8 , 16 , 17 ] for more information.…”

“…Assume now that and is a sequence of elementary functions which satisfies in the following form then, the Itô integral of g [ 17 ] is defined as follows: …”

“…Such as: operational matrix method based on hat functions [13] and block-pulse [14]. Also, some studies have been performed on different types of stochastic differential integral equations [15][16][17].…”