Automatic solution of integral equations pertinent to diffusion with first order homogeneous reactions at cylindrical wire electrodes

SynopsisThough it is still an open problem for which class of first-order elliptic systems Carleman's theorem holds, this is proven here for a certain class of systems (with analytic coefficients) for which Douglis introduced the hypercomplex algebra and hyperanalytic functions. The proof is based on a representation formula generalising Vekua's approach with Volterra integral equations in C2 to more than two unknowns. The representation formula is of its own interest because it provides the generation of complete families of solutions. The equations of plane inhomogeneous elasticity problems lead to a system of the desired class.

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Numerical solution of the system of Marchenko integral equations

Uzbek Mathematical Journal ◽

10.29229/uzmj.2021-3-16 ◽

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.

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Linear first order Riemann-Liouville fractional differential and perturbed Abel's integral equations

Journal of Differential Equations ◽

10.1016/j.jde.2021.10.025 ◽

2022 ◽

Vol 306 ◽

pp. 28-59

Author(s):

Kunquan Lan

Keyword(s):

Integral Equations ◽

First Order ◽

Fractional Differential

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On a class of systems of singular integral equations on a bounded domain

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500011641 ◽

1979 ◽

Vol 83 (3-4) ◽

pp. 347-364 ◽

Cited By ~ 3

Author(s):

A. Džuraev

Keyword(s):

Integral Equations ◽

Singular Integral ◽

Singular Integral Equations ◽

Boundary Curve ◽

Adjoint Problem ◽

Bergman Kernel Function ◽

One Dimensional ◽

First Order ◽

Order Elliptic System ◽

Type Boundary

SynopsisA class of systems of two-dimensional singular integral equations with even kernels over bounded domains in the plane is studied. The applications include integral equations with the Bergman kernel function. The method of investigation is the following: the integral equation is reduced to a Riemann type boundary value problem for a first order elliptic system. This is solved by means of one-dimensional singular integral equations over the boundary curve. An adjoint problem is formulated, the Noetherian theorems are established, and a formula for the index is given.

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Nonlinear impulsive Volterra integral equations in Banach spaces and applications

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953393000048 ◽

1993 ◽

Vol 6 (1) ◽

pp. 35-48 ◽

Cited By ~ 12

Author(s):

Dajun Guo

Keyword(s):

Differential Equations ◽

Banach Spaces ◽

Integral Equations ◽

Initial Value Problems ◽

Infinite System ◽

Impulsive Differential Equations ◽

Volterra Integral Equations ◽

Initial Value ◽

First Order ◽

Maximal Solutions

In this paper, we first extend results on the existence of maximal solutions for nonlinear Volterra integral equations in Banach spaces to impulsive Volterra integral equations. Then, we give some applications to initial value problems for first order impulsive differential equations in Banach spaces. The results are demonstrated by means of an example of an infinite system for impulsive differential equations.

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