scholarly journals On an Inverse Problem of Reconstructing a Heat Conduction Process from Nonlocal Data

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Makhmud A. Sadybekov ◽  
Gulnar Dildabek ◽  
Marina B. Ivanova
Keyword(s):  
Inverse Problem ◽  
Space Variable ◽  
Separation Of Variables ◽  
Heat Propagation ◽  
Final States ◽  
One Dimensional ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
The Given ◽  
Heat Conduction Process

We consider an inverse problem for a one-dimensional heat equation with involution and with periodic boundary conditions with respect to a space variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for redefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

scholarly journals A STUDY OF NONLINEAR FRACTIONAL-ORDER BOUNDARY VALUE PROBLEM WITH NONLOCAL ERDELYI-KOBER AND GENERALIZED RIEMANN-LIOUVILLE TYPE INTEGRAL BOUNDARY CONDITIONS

2017 ◽  
Vol 22 (2) ◽  
pp. 121-139 ◽  
Author(s):  
Bashir Ahmad ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon ◽  
Ahmed Alsaedi
Keyword(s):  
Boundary Conditions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Fractional Integrals ◽  
Integral Boundary ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Caputo Fractional Differential Equations ◽  
The Given

We investigate a new kind of nonlocal boundary value problems of nonlinear Caputo fractional differential equations supplemented with integral boundary conditions involving Erdelyi-Kober and generalized Riemann-Liouville fractional integrals. Existence and uniqueness results for the given problem are obtained by means of standard fixed point theorems. Examples illustrating the main results are also discussed.

scholarly journals ON EXISTENCE OF SOLUTIONS FOR NONLINEAR Q-DIFFERENCE EQUATIONS WITH NONLOCAL Q-INTEGRAL BOUNDARY CONDITIONS

2015 ◽  
Vol 20 (5) ◽  
pp. 604-618 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Guotao Wang ◽  
Bashir Ahmad ◽  
Lihong Zhang ◽  
Aatef Hobiny ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Existence Of Solutions ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Point Theory ◽  
Integral Boundary ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
The Given

In this paper, we discuss the existence of solutions for nonlinear qdifference equations with nonlocal q-integral boundary conditions. The first part of the paper deals with some existence and uniqueness results obtained by means of standard tools of fixed point theory. In the second part, sufficient conditions for the existence of extremal solutions for the given problem are established. The results are well illustrated with the aid of examples.

Existence and uniqueness results for an inverse problem for semilinear parabolic equations

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1057-1064 ◽  
Author(s):  
Ali Sazaklioglu ◽  
Allaberen Ashyralyev ◽  
Abdullah Erdogan
Keyword(s):  
Inverse Problem ◽  
Difference Scheme ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Semilinear Parabolic Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Accuracy Difference ◽  
Integral Overdetermination ◽  
Semilinear Parabolic

In the present study, unique solvability of an inverse problem governed by semilinear parabolic equations with an integral overdetermination is investigated. Furthermore, for the approximate solution of this problem a first order of accuracy difference scheme is constructed. Existence and uniqueness results for the solution of this difference scheme are established. Considering a particular example, some numerical results are discussed.

scholarly journals Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions

2019 ◽  
Vol 3 (2) ◽  
pp. 34 ◽  
Author(s):  
Ahmad ◽  
Alsaedi ◽  
Salem ◽  
Ntouyas
Keyword(s):  
Fixed Point Theory ◽  
Point Theory ◽  
Stieltjes Integral ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results ◽  
Special Cases ◽  
Mixed Nonlinearities ◽  
Fractional Differential ◽  
The Given

In this paper, we investigate a new class of boundary value problems involving fractionaldifferential equations with mixed nonlinearities, and nonlocal multi-point and Riemann–Stieltjesintegral-multi-strip boundary conditions. Based on the standard tools of the fixed point theory,we obtain some existence and uniqueness results for the problem at hand, which are well illustratedwith the aid of examples. Our results are not only in the given configuration but also yield severalnew results as special cases. Some variants of the given problem are also discussed.

scholarly journals Existence and uniqueness results for an inverse problem for a semilinear equation with final overdetermination

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 847-858 ◽  
Author(s):  
Ali Sazaklioglu ◽  
Abdullah Erdogan ◽  
Allaberen Ashyralyev
Keyword(s):  
Banach Space ◽  
Inverse Problem ◽  
Difference Scheme ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Order Of Accuracy ◽  
First Order ◽  
Existence And Uniqueness Results ◽  
Semilinear Equation ◽  
Uniqueness Results

In the present paper, unique solvability of a source identification inverse problem for a semilinear equation with a final overdetermination in a Banach space is investigated. Moreover, the first order of accuracy Rothe difference scheme is presented for numerically solving this problem. The existence and uniqueness result for this difference scheme is given. The efficiency of the proposed method is evaluated by means of computational experiments.

A biodegradable elastic stent model

2018 ◽  
Vol 24 (8) ◽  
pp. 2591-2618
Author(s):  
Josip Tambača ◽  
Bojan Žugec
Keyword(s):  
Cross Sections ◽  
Applied Mathematics ◽  
Contact Forces ◽  
Dimensional Model ◽  
One Dimensional ◽  
Vascular Stents ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Curved Rods ◽  
The One

In this paper we derive and analyse a one-dimensional model of biodegradable elastic stents. The model is given as a nonlinear system of ordinary differential equations on a graph defined by the geometry of stent struts. The unknowns in the problem are the displacement of the middle curve of the struts, the infinitesimal rotation of the cross-sections of the stent struts, the contact couples and contact forces at struts and a function describing the degradation of the stent. The model is based on the one-dimensional model of a biodegradable elastic curved rod model by Tambača and Žugec (‘One-dimensional quasistatic model of biodegradable elastic curved rods’, Zeitschrift für Angewandte Mathematik und Physik 2015; 66(5): 2759–2785) and the ideas from the one-dimensional elastic stent modelling by Tambača et al. (‘Mathematical modeling of vascular stents’, SIAM Journal on Applied Mathematics 2010; 70(6): 1922–1952) used to formulate contact conditions at vertices. We prove the existence and uniqueness results for the model.

scholarly journals An inverse problem for one-dimensional diffusion equation in optical tomography

2017 ◽  
Vol 37 (3) ◽  
pp. 159-194
Author(s):  
Mohamed Addam
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Optical Tomography ◽  
Approximate Solutions ◽  
Photon Density ◽  
Optical Parameters ◽  
Good Resolution ◽  
Forward Solution ◽  
One Dimensional ◽  
Uniqueness Results

In this paper, we study the one-dimensional inverse problem for the diffusion equation based optical tomography. The objective of the present work is a mathematical and numerical analysis concerning one-dimensional inverse problem. In the first stage, the forward diffusion equation with boundary conditions is solved using an intermediate elliptic equation. We give the existence and the uniqueness results of the solution. An approximation of the photon density in frequency-domain is proposed using a Splines Galerkin method. In the second stage, we give theoretical results such as the stability and lipschitz-continuity of the forward solution and the Fréchet differentiability of the Dirichlet-to-Neumann nonlinear map with respect to the optical parameters. The Fréchet derivative is used to linearize the considered inverse problem. The Newton method based on the regularization technique will allow us to compute the approximate solutions of the inverse problem. Several test examples are used to verify high accuracy, effectiveness and good resolution properties for smooth and discontinuous optical property solutions.

scholarly journals A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions

2022 ◽  
Vol 6 (1) ◽  
pp. 45
Author(s):  
Ravi P. Agarwal ◽  
Hana Al-Hutami ◽  
Bashir Ahmad
Keyword(s):  
Boundary Conditions ◽  
Difference Equations ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results ◽  
Multipoint Boundary Conditions ◽  
The Given ◽  
Variant System

We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the given problem. Illustrative examples for the obtained results are also presented.

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