We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose solutions exactly coincide with solutions of the corresponding differential equation evaluated at a discrete sequence of points (a lattice). Such exact discret...

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...

In this work first we consider a physiologically structured population model with a distributed recruitment process. That is, our model allows newly recruited individuals to enter the population at all possible individual states, in principle. The model can be naturally formulated as a first order partial integro-differential equation, and it has been studied extensively. In particular, it is w...

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 will prove the well-posedness of certain second order ordinary and partial integro-differential equations with an integral term of the form t ∫ 0 m(φ(t−s))φ̇(s) ds, wherem is given and φ is the solution. The new aspect is the dependence of the kernel on the solution. In addition, for the ordinary integro-differential equation, the asymptotic behaviour of the solution is described for some ker...

This paper is concerned with the valuation of options in jump diffusion models. The partial integro-differential equation (PIDE) inherent in the pricing problem is solved by using the Mellin integral transform. The solution is a single integral expression independent of the distribution of the jump size. We also derive analytical expressions for the Greeks. The results are implemented and compa...

In this paper a higher-order numerical solution of a non-linear Volterra integro-differential equation is discussed. Example of this question has been solved numerically using the Runge-Kutta-Verner method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulas for integral parts.

In a di erential equations course, one typically studies a few classes of problems for which there are closed form solutions, such as ordinary, linear di erential equations with constant coe cients. Most problems of interest in real world simulation problems are much more complex, however, involving domains of two or more dimensions or nonlinear e ects, yielding partial di erential equations or...

In this paper we study an integro-differential equation describing granular flow dynamics with slow erosion. This nonlinear partial differential equation is a conservation law where the flux contains an integral term. Through a generalized wave front tracking algorithm, approximate solutions are constructed and shown to converge strongly to a Lipschitz semigroup.

A nonlinear integro-differential equation of convolution type with order of nonlinearity more than one and a stable trivial solution is considered. The integral in this equation has an exponential kernel and polynomial integrand. The difference analogue of the equation considered is constructed in the form of a difference equation with continuous time and it is shown that this difference analog...