We consider pricing options in a jump-diffusion model which requires solving a partial integro-differential equation. Discretizing the spatial direction with a fourth order compact scheme leads to a linear system of ordinary differential equations. For the temporal direction, we utilize the favorable boundary value methods owing to their advantageous stability properties. In addition, the resul...

In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of a non-autonomous integro-partial differential equation describing the heat flow in a rigid heat conductor with memory Existence and uniqueness of solutions is provided. Moreover, under proper assumptions on the heat flux memory kernel and on the magnitude of nonlinearity, the existence of unifor...

In this paper we consider a risk model with two classes of insurance risks in the presence of multiple thresholds. We assume that the two claim counting processes are, respectively, Poisson and Sparre Andersen with generalized Erlang(2) claim inter-arrival times. We derive an integro-differential system for the Gerber-Shiu functions for surplus-dependent premium rates and a piecewise integro-di...

In this paper, we prove the existence and uniqueness of a nonlinear perturbed stochastic fractional integro-differential equation of Volterra-Itô type involving nonlocal initial condition by using the theory of admissibility of integral operator and Banach fixed-point principle. Also the stability and boundedness of the second moments of the stochastic solution are studied. In addition, an appl...

A finite element method formulation is developed for the solution of anomalous diffusion equation. This equation belongs to branch mathematics called fractional calculus: it governed by a partial differential in which time derivative, whose order ranges interval (0,1), replaces first-order derivative classical In this work, Caputo integro-differential operator employed represent derivative. Aft...

In this paper, we study the problem of solving Seal’s type partial integro-differential equations (PIDEs) for classical compound Poisson risk model. A data-driven deep neural network (DNN) method is proposed to calculate finite-time survival probability, and an alternative scheme also investigated when claim payments are exponentially distributed. The DNN then extended numerical solution genera...

This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation (PIDE) with a weakly singular kernel. Cubic trigonometric B-spline (CTBS) functions are used interpolation in both methods. The first method is CTBS based collocation which reduces PIDE to an algebraic tridiagonal system linear equations. other differentia...

this paper has been devoted to apply the reconstruction of variational iteration method (rvim) to handle the systems of integro-differential equations. rvim has been induced with laplace transform from the variational iteration method (vim) which was developed from the inokuti method. actually, rvim overcome to shortcoming of vim method to determine the lagrange multiplier. so that, rvim method...

in this paper, we propose the chebyshev wavelet approximation for the numerical solution of a class of integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields. we show that the chebyshev approximation transform an integral equation to an explicit system of linear algebraic equations. illustrative examples are included t...