Numerical Solution of Nonlinear Stochastic Itô–Volterra Integral Equations Driven by Fractional Brownian Motion Using Block Pulse Functions

This paper presents a valid numerical method to solve nonlinear stochastic Itô–Volterra integral equations (SIVIEs) driven by fractional Brownian motion (FBM) with Hurst parameter H ∈ 1 / 2 , 1 . On the basis of FBM and block pulse functions (BPFs), a new stochastic operational matrix is proposed. The nonlinear stochastic integral equation is converted into a nonlinear algebraic equation by this method. Furthermore, error analysis is given by the pathwise approach. Finally, two numerical examples exhibit the validity and accuracy of the approach.

Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations

The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the use of IBPFs in addition to the operational matrix of integration. Typically, the classical alterations have enhanced the time taken by the computer program to solve the system of algebraic equations. The current modification works perfectly and has improved the efficiency over the regular block–pulse basis functions (BPF). Additionally, the paper handles the uniqueness plus the convergence theorems of the solution. Numerical examples have been presented to illustrate the efficiency as well as the accuracy of the method. Furthermore, tables and graphs are used to show and confirm how the method is highly efficient.

Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

NUMERICAL SOLUTION OF MIXED LINEAR VOLTERRA-FREDHOLM INTEGRAL EQUATIONS BY MODIFIED BLOCK PULSE FUNCTIONS

A numerical method based on modified block pulse functions is proposed for solving the mixed linear Volterra-Fredholm integral equations. We obtain an integration operational matrix of modified block pulse functions on interval [0,T). A modified block pulse functions and their operational matrix of integration, the mixed linear Volterra-Fredholm integral equations can be reduced to a linear system of algebraic equations. The rate of convergence is O(h) and error analysis of the proposed method are discussed. Some examples are provided to show that the proposed method have a good degree of accuracy.