An explicit numerical algorithm to the solution of Volterra integral equation of the second kind

08/07/2019
by   Leanne Dong, et al.
0

This paper considers a numeric algorithm to solve the equation y(t)=f(t)+∫^t_0 g(t-τ)y(τ) dτ with a kernel g and input f for y. In some applications we have a smooth integrable kernel but the input f could be a generalised function, which could involve the Dirac distribution. We call the case when f=δ, the Dirac distribution centred at 0, the fundamental solution E, and show that E=δ+h where h is integrable and solve h(t)=g(t)+∫^t_0 g(t-τ)h(τ) dτ The solution of the general case is then y(t)=f(t)+(h*f)(t) which involves the convolution of h and f. We can approximate g to desired accuracy with piecewise constant kernel for which the solution h is known explicitly. We supply an algorithm for the solution of the integral equation with specified accuracy.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/30/2022

Fast Computation of Electrostatic Potentials for Piecewise Constant Conductivities

We present a novel numerical method for solving the elliptic partial dif...
research
09/09/2019

On the solutions of linear Volterra equations of the second kind with sum kernels

We consider a linear Volterra integral equation of the second kind with ...
research
09/14/2023

The kernel-balanced equation for deep neural networks

Deep neural networks have shown many fruitful applications in this decad...
research
06/03/2013

Constructive Setting of the Density Ratio Estimation Problem and its Rigorous Solution

We introduce a general constructive setting of the density ratio estimat...
research
10/23/2018

Fast Computation of Steady-State Response for Nonlinear Vibrations of High-Degree-of-Freedom Systems

We discuss an integral equation approach that enables fast computation o...
research
11/15/2021

A note on averaging prediction accuracy, Green's functions and other kernels

We present the mathematical context of the predictive accuracy index and...
research
07/22/2020

Numerical solution of a one-dimensional nonlocal Helmholtz equation by Perfectly Matched Layers

We consider the computation of a nonlocal Helmholtz equation by using Pe...

Please sign up or login with your details

Forgot password? Click here to reset