We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite family standard SDEs and establish its well-posedness, show that it is equivalent to equation. also representation formula in terms non-linear semi-linear partial differential equation Hamilton–Jacobi–Bellman type. As application, consider the tim...

We construct the basis of a stochastic calculus for so-called Volterra processes, i.e., processes which are defined as the stochastic integral of a timedependent kernel with respect to a standard Brownian motion. For these processes which are natural generalization of fractional Brownian motion, we construct a stochastic integral and show some of its main properties: regularity with respect to ...

In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of second kind with a weakly singular kernel. We use some function transformation and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi or...

In this paper, we continue our study that began in recent papers [2] and [3] concerning a simple yet effective Taylor series expansion method to approximate a solution of integral equations. The method is applied to Volterra integral equation of the second kind as well as to systems of Volterra equations. The results obtained in this paper improve significantly the results reported in recent pa...

We consider numerical solutions of a class of nonlinear (nonstandard) Volterra integral equations. We first prove the existence and uniqueness of the solution of the Volterra integral equation in the context of the space of continuous funtions over a closed interval. We then use one point collocation methods and quadrature methods with a uniform mesh to construct solutions of the nonlinear VIE....

In this work we are concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation with a nonsmooth solution. We investigate the application of product integration methods and a detailed analysis of the Trapezoidal method is given. In order to improve the numerical results we consider extrapolation procedures and collocation methods based on graded meshes. Sever...

in this paper, we introduce an efficient method based on haar wavelet to approximate a solutionfor the two-dimensional linear stochastic fredholm integral equation. we also give an example to demonstrate the accuracy of the method.

it is well known that the parabolic partial differential equationsin two or more space dimensions with overspecified boundary data,feature in the mathematical modeling of many phenomena. in thisarticle, an inverse problem of determining an unknowntime-dependent source term of a parabolic equation in generaldimensions is considered. employing some transformations, wechange the inverse problem to...