Bilinear approach is applied to derive integrable multi-component generalizations of the socalled 1+1 dimensional special Toda lattice, the Volterra lattice, a simple differential-difference equation found by Adler, Moser, Weiss, Veselov and Shabat and another integrable lattice reduced from the discrete BKP equation. Their soliton solutions expressed by pfaffians and the corresponding bilinear...

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.

We prove that a set of weak solutions of the nonlinear Volterra integral equation has the Kneser property. The main condition in our result is formulated in terms of axiomatic measures of weak noncompactness.

Direct inversion of acoustic scattering problems is nonlinear. One way to treat the inverse scattering problem is based on the reversion of the Born–Neumann series solution of the Lippmann–Schwinger equation. An important issue for this approach is the radius of convergence of the Born–Neumann series for the forward problem. However, this issue can be tackled by employing a renormalization tech...

The most robust treatment of the inverse acoustic scattering problem is based on the reversion of the Born-Neumann series solution of the Lippmann-Schwinger equation. An important issue for this approach to inversion is the radius of convergence of the Born-Neumann series for Fredholm integral kernels, and especially for acoustic scattering for which the interaction depends on the square of the...

In this paper, a nonlinear Volterra-Fredholm integral equation of the first kind is solved by using the homotopy analysis method (HAM). In this case, the first kind integral equation can be reduced to the second kind integral equation which can be solved by HAM. The approximate solution of this equation is calculated in the form of a series which its components are computed easily. The accuracy...

Abstract: A one dimensional heat equation in a semi-infinite medium controlled through a heat source depending on the delayed heat flux at the extremum is studied. By reducing the problem to a delayed Volterra integral equation with a weakly singular kernel, we find conditions on the initial datum and on the source term of the equation to control the asymptotic behaviour of the mean temperatures.