The experimental results of Saffman & Taylor (1958) and Pith (1980) on fingering in a Hele Shaw cell are modelled by two-dimensional potential flow with surface-tension effects included at the interface. Using free streamline techniques, the shape of the free surface is expressed rn the solution of a nonlinear integro-differential equation. The equation is solved numerically and the solutions a...

The KP equation, which is an integrable nonlinear evolution equation in 2 + 1, i.e., two spatial and one temporal dimensions, is a physically significant generalization of the KdV equation. The question of constructing an integrable generalization of the KP equation in 3+1, has been one of the central open problems in the field of integrability. By complexifying the independent variables of the...

the spline collocation method  is employed to solve a system of linear and nonlinear fredholm and volterra integro-differential equations. the solutions are collocated by cubic b-spline and the integrand is approximated by the newton-cotes formula. we obtain the unique solution for linear and nonlinear system $(nn+3n)times(nn+3n)$ of integro-differential equations. this approximation reduces th...

In this paper, Itô stochastic integro-differential equations are considered. By establishing an L -operator integro-differential inequality and using the properties of M -cone and stochastic analysis technique, we obtain some new sufficient conditions ensuring the exponential p -dissipativity of the stochastic integro-differential equations. An example is also discussed to illustrate the effici...

Integro-differential polynomials are a novel generalization of the well-known differential polynomials extensively used in differential algebra [17]. They were introduced in [29] as a kind of universal extensions of integro-differential algebras and have recently been applied in a confluence proof [34] for the rewrite system underlying the so-called “integro-differential operators”. In this pap...

In our symbolic approach to boundary problems for linear ordinary differential equations we use the algebra of integro-differential operators as an algebraic analogue of differential, integral and boundary operators (Section 2). They allow to express the problem statement (differential equation and boundary conditions) as well as the solution operator (an integral operator called “Green’s opera...