Consider a stochastic difference equation xi+1 = Gi [xi, xi−1, . . . , x0] + fi [xi, xi−1, . . . , x0] + k0 ∑ k=0 σi−k [xi−k, xi−k−1, . . . , x0] ξi+1−k, (1) with the Volterra type nonlinear main term G and the Volterra type noise f + ∑k0 k=0 σi−kξi+1−k. Functions Gi, fi, σi are random while ξi is a martingale-difference. In general, k0 ≥ 1 so Eqn (1) cannot be considered as a stochastic equati...

Keywords: Volterra–Stieltjes integral equation Fractional integral–differential equations Riemann–Liouville fractional operators Existence and stability of solutions Fixed point a b s t r a c t Our aim in this paper is to study the existence and the stability of solutions for Riemann–Liouville Volterra–Stieltjes quadratic integral equations of fractional order. Our results are obtained by using...

In this article, we deal with fractional stochastic differential equations, so-called Caputo type backward equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to fBSDEs order $$\alpha \in (\frac{1}{2},1)$$ whose coefficients satisfy a Lipschitz condition. A novelty article is that introduce new weighted norm in square integrable measurable function space us...

Since various problems in science and engineering fields can be modeled by nonlinear Volterra-Fredholm integral equations, the main focus of this study is to present an effective numerical method for solving them. This method is based on the hybrid functions of Legendre polynomials and block-pulse functions. By using this approach, a nonlinear Volterra-Fredholm integral equation reduces to a no...

The Fredholm–Volterra integral equation of the second kind with continuous kernels with respect to position and time, is solved numerically, using the Collocation and Galerkin methods. Also the error, in each case, is estimated.

M. A. Abdou, Khamis I. Mohamed and A. S. Ismail, On the numerical solutions of Fredholm-Volterra integral equation, Appl. Math. Comp. 146, 713-728, (2003). M. A. Abdou, Khamis I. Mohamed and A. S. Ismail, Toeplitz Matrix and product Nystrom methods for solving the singular integral equation, Le Matematiche LVII-Fasc. I, 21-37, (2002). H. Brunner, On the numerical solution of nonlinear VolterraF...

Homotopy perturbation method is a well-known method for solving many functional equations such as differential equations, integral equations, integro-differential equations and so on. Applying this method needs some computations which are boring by hand. Therefore, creating a program to do all computations would be useful. In this work, a maple program is prepared to solve the second kind of Vo...

We prove the existence and uniqueness, local in time, of a solution for a one-phase Stefan problem of a non-classical heat equation for a semiinfinite material with temperature boundary condition at the fixed face. We use the Friedman-Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations.

In this paper, we study a class of Type-II backward stochastic Volterra integral equations (BSVIEs). For the adapted M-solutions, obtain two approximation results, namely, BSDE and numerical approximation. The means that solution finite system differential (BSDEs) converges to M-solution original equation. As consequence approximation, an estimate for L 2 -time regularity M-solutions BSVIEs. pr...