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11 pages, 873 KiB

Open AccessArticle

Solution of the Vector Three-Dimensional Inverse Problem on an Inhomogeneous Dielectric Hemisphere Using a Two-Step Method

by Eugen Smolkin, Yury Smirnov and Maxim Snegur

Computation 2024, 12(11), 213; https://doi.org/10.3390/computation12110213 - 22 Oct 2024

Viewed by 964

Abstract

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for [...] Read more.

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for Maxwell’s equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography. Full article

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12 pages, 292 KiB

Open AccessArticle

The Ulam Stability of High-Order Variable-Order φ-Hilfer Fractional Implicit Integro-Differential Equations

by Peiguang Wang, Bing Han and Junyan Bao

Fractal Fract. 2024, 8(9), 502; https://doi.org/10.3390/fractalfract8090502 - 26 Aug 2024

Viewed by 1057

Abstract

This study investigates the initial value problem of high-order variable-order

φ

-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional [...] Read more.

This study investigates the initial value problem of high-order variable-order

φ

-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions. Full article

(This article belongs to the Section General Mathematics, Analysis)

24 pages, 1357 KiB

Open AccessArticle

Solving a System of Caputo Fractional-Order Volterra Integro-Differential Equations with Variable Coefficients Based on the Finite Difference Approximation via the Block-by-Block Method

by Shazad Shawki Ahmed and Shokhan Ahmed Hamasalih

Symmetry 2023, 15(3), 607; https://doi.org/10.3390/sym15030607 - 27 Feb 2023

Cited by 3 | Viewed by 1660

Abstract

This paper focuses on computational technique to solve linear systems of Volterra integro-fractional differential equations (LSVIFDEs) in the Caputo sense for all fractional order

lins in (0, 1]

using two and three order block-by-block approach with explicit finite difference approximation. With this [...] Read more.

This paper focuses on computational technique to solve linear systems of Volterra integro-fractional differential equations (LSVIFDEs) in the Caputo sense for all fractional order

lins in (0, 1]

using two and three order block-by-block approach with explicit finite difference approximation. With this method, we aim to use an appropriate process to transform our problem into an analogous piecewise iterative linear algebraic system. Moreover, algorithms for treating LSVIFDEs using the above process have been developed, in order to express these solutions. In addition, numerical examples for our scheme are presented based on various kernels, including symmetry kernel and other sorts of separate kernels, are used to illustrate the validity, effectiveness and applicability of the suggested approach. Consequently, comparisons are performed with exact results using this technique, to communicate these approaches most general programs are written in Python V 3.8.8 software

(2021)

. Full article

(This article belongs to the Section Mathematics)

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19 pages, 309 KiB

Open AccessArticle

Piecewise Fractional Jacobi Polynomial Approximations for Volterra Integro-Differential Equations with Weakly Singular Kernels

by Haiyang Li and Junjie Ma

Axioms 2022, 11(10), 530; https://doi.org/10.3390/axioms11100530 - 4 Oct 2022

Cited by 2 | Viewed by 2017

Abstract

This paper is concerned with numerical solutions to Volterra integro-differential equations with weakly singular kernels. Making use of the transformed fractional Jacobi polynomials, we develop a class of piecewise fractional Galerkin methods for solving this kind of Volterra equation. Then, we study the [...] Read more.

This paper is concerned with numerical solutions to Volterra integro-differential equations with weakly singular kernels. Making use of the transformed fractional Jacobi polynomials, we develop a class of piecewise fractional Galerkin methods for solving this kind of Volterra equation. Then, we study the existence, uniqueness and convergence properties of Galerkin solutions by exploiting the decaying rate of the coefficients of the transformed fractional Jacobi series. Finally, numerical experiments are carried out to illustrate the performance of the piecewise Galerkin solution. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)

15 pages, 330 KiB

Open AccessArticle

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

by Abdelmonaim Saou, Driss Sbibih, Mohamed Tahrichi and Domingo Barrera

Mathematics 2022, 10(6), 893; https://doi.org/10.3390/math10060893 - 11 Mar 2022

Cited by 2 | Viewed by 1957

Abstract

The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By [...] Read more.

The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree

⩽ r - 1

, we obtain convergence order

2 r

for degenerate kernel and Nyström methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are

3 r + 1

and

4 r

, respectively. Moreover, we show that the optimal convergence order

4 r

is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples. Full article

(This article belongs to the Special Issue Spline Functions and Applications)

18 pages, 325 KiB

Open AccessArticle

On a Fractional Stochastic Risk Model with a Random Initial Surplus and a Multi-Layer Strategy

by Enrica Pirozzi

Mathematics 2022, 10(4), 570; https://doi.org/10.3390/math10040570 - 12 Feb 2022

Cited by 2 | Viewed by 1749

Abstract

The paper deals with a fractional time-changed stochastic risk model, including stochastic premiums, dividends and also a stochastic initial surplus as a capital derived from a previous investment. The inverse of a

ν

-stable subordinator is used for the time-change. The submartingale property [...] Read more.

The paper deals with a fractional time-changed stochastic risk model, including stochastic premiums, dividends and also a stochastic initial surplus as a capital derived from a previous investment. The inverse of a

ν

-stable subordinator is used for the time-change. The submartingale property is assumed to guarantee the net-profit condition. The long-range dependence behavior is proven. The infinite-horizon ruin probability, a specialized version of the Gerber–Shiu function, is considered and investigated. In particular, we prove that the distribution function of the infinite-horizon ruin time satisfies an integral-differential equation. The case of the dividends paid according to a multi-layer dividend strategy is also considered. Full article

(This article belongs to the Special Issue Stochastic Models with Applications)

23 pages, 7469 KiB

Open AccessArticle

A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D

by Gujji Murali Mohan Reddy, Alan B. Seitenfuss, Débora de Oliveira Medeiros, Luca Meacci, Milton Assunção and Michael Vynnycky

Algorithms 2020, 13(10), 242; https://doi.org/10.3390/a13100242 - 24 Sep 2020

Cited by 4 | Viewed by 3692

Abstract

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the [...] Read more.

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs. Full article

(This article belongs to the Section Algorithms for Multidisciplinary Applications)

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20 pages, 947 KiB

Open AccessFeature PaperArticle

On Computations in Renewal Risk Models—Analytical and Statistical Aspects

by Josef Anton Strini and Stefan Thonhauser

Risks 2020, 8(1), 24; https://doi.org/10.3390/risks8010024 - 4 Mar 2020

Cited by 3 | Viewed by 3490

Abstract

We discuss aspects of numerical methods for the computation of Gerber-Shiu or discounted penalty-functions in renewal risk models. We take an analytical point of view and link this function to a partial-integro-differential equation and propose a numerical method for its solution. We show [...] Read more.

We discuss aspects of numerical methods for the computation of Gerber-Shiu or discounted penalty-functions in renewal risk models. We take an analytical point of view and link this function to a partial-integro-differential equation and propose a numerical method for its solution. We show weak convergence of an approximating sequence of piecewise-deterministic Markov processes (PDMPs) for deriving the convergence of the procedures. We will use estimated PDMP characteristics in a subsequent step from simulated sample data and study its effect on the numerically computed Gerber-Shiu functions. It can be seen that the main source of instability stems from the hazard rate estimator. Interestingly, results obtained using MC methods are hardly affected by estimation. Full article

(This article belongs to the Special Issue Loss Models: From Theory to Applications)

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