A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

We present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

Application of Jacobi Polynomial and Multivariable Aleph- Function in Heat Conduction in Non-Homogeneous Moving Rectangular Parallelepiped

The present paper deals with an application of Jacobi polynomial and multivariable Aleph-function to solve the differential equation of heat conduction in non-homogeneous moving rectangular parallelepiped. The temperature distribution in the parallelepiped, moving in a direction of the length (x-axis) between the limits x = −1 and x = 1 has been considered. The conductivity and the velocity have been assumed to be variables. We shall see two particular cases and the cases concerning Aleph-function of two variables and the I-function of two variables.

Exact scattering and bound states solutions for novel hyperbolic potentials with inverse square singularity

We use the Tridiagonal Representation Approach (TRA) to obtain exact scattering and bound states solutions of the Schrödinger equation for short-range inverse-square singular hyperbolic potentials. The solutions are series of square integrable functions written in terms of the Jacobi polynomial with the Wilson polynomial as expansion coefficients. The series is finite for the discrete bound states and infinite but bounded for the continuum scattering states.

An Efficient Numerical Method for Solving Volterra-Fredholm Integro-Differential Equations of Fractional Order by Using Shifted Jacobi-Spectral Collocation Method

The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.