scholarly journals Iterative Splitting Methods for Integrodifferential Equations: Theory and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Jürgen Geiser
Keyword(s):  
Differential Equations ◽  
Splitting Methods ◽  
Integrodifferential Equations ◽  
Cauchy Problems ◽  
Splitting Schemes ◽  
Scattering Problems ◽  
First Order ◽  
Fast Solver ◽  
Transport Problems ◽  
Abstract Cauchy Problems

We present novel iterative splitting methods to solve integrodifferential equations. Such integrodifferential equations are applied, for example, in scattering problems of plasma simulations. We concentrate on a linearised integral part and a reformulation to a system of first order differential equations. Such modifications allow for applying standard iterative splitting schemes and for extending the schemes, respecting the integral operator. A numerical analysis is presented of the system of semidiscretised differential equations as abstract Cauchy problems. In the applications, we present benchmark and initial realistic applications to transport problems with scattering terms. We also discuss the benefits of such iterative schemes as fast solver methods.

scholarly journals Fast methods for the solution of singular integro-differential and differential equations

1981 ◽  
Vol 4 (4) ◽  
pp. 775-794
Author(s):  
L. F. Abd-Elal
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Integrodifferential Equations ◽  
Solution Time ◽  
First Order ◽  
Chebyshev Expansion ◽  
Fast Methods ◽  
The Galerkin Method

Uniform methods based on the use of the Galerkin method and different Chebyshev expansion sets are developed for the numerical solution of linear integrodifferential equations of the first order. These methods take a total solution time0(N2lnN)usingNexpansion functions, and also provide error extimates which are cheap to compute. These methods solve both singular and regular integro-differential equations. The methods are also used in solving differential equations.

Numerical solution of the system of Marchenko integral equations

2021 ◽  
Vol 65 (3) ◽  
pp. 159-165
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Algebraic Equations ◽  
System Of Differential Equations ◽  
Scattering Problems ◽  
First Order ◽  
First Order Differential Equations

In this paper, inverse scattering problems for a system of differential equations of the first order are considered. The Marchenko approach is used to solve the inverse scattering problem. The system of Marchenko integral equations is reduced to a linear system of algebraic equations such that the solution of the resulting system yields to the unknown coefficients of the system of first-order differential equations. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.

scholarly journals Propagation relations for solutions of some higher order cauchy problems

1996 ◽  
Vol 38 (2) ◽  
pp. 229-239 ◽  
Author(s):  
L. R. Bragg
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Solution Properties ◽  
Cauchy Problems ◽  
Wave Problem ◽  
Basic Algorithm ◽  
Solution Behaviour ◽  
Abstract Cauchy Problems ◽  
Addition Formulas

AbstractThe Huygens' property is exploited to study propagation relations for solutions of certain types of linear higher order Cauchy problems. Motivated by the solution properties of the abstract wave problem, addition formulas are developed for the solution operators of these problems. The application of these alternative forms of the solution operators to data leads to connecting operator relations between distinct solutions of the problems at different times. We examine this solution behaviour for both analytic and abstract Cauchy problems. A basic algorithm for constructing addition formulas for solutions of ordinary differential equations is included.

scholarly journals The generalized Wiener–Hopf equations for wave motion in angular regions: electromagnetic application

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210040
Author(s):  
V. G. Daniele ◽  
G. Lombardi
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Current Literature ◽  
Functional Equations ◽  
Wave Motion ◽  
Scattering Problems ◽  
First Order ◽  
Homogeneous Media ◽  
General Method

In this work, we introduce a general method to deduce spectral functional equations and, thus, the generalized Wiener–Hopf equations (GWHEs) for wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity with application to electromagnetics. The functional equations are obtained by solving vector differential equations of first order that model the problem. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper shows the general theory and the validity of GWHEs in the context of electromagnetic applications with respect to the current literature. Extension to scattering problems by wedges in arbitrarily linear media in different physics will be presented in future works.

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