scholarly journals Inverse boundary-value problem for the equation of longitudinal wave propagation with non-self-adjoint boundary conditions

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5259-5271
Author(s):  
Elvin Azizbayov ◽  
Yashar Mehraliyev
Keyword(s):  
Inverse Problem ◽  
Wave Propagation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Longitudinal Wave ◽  
Existence And Uniqueness ◽  
Original Problem ◽  
Boundary Value ◽  
Inverse Boundary ◽  
Longitudinal Wave Propagation

We study the inverse coefficient problem for the equation of longitudinal wave propagation with non-self-adjoint boundary conditions. The main purpose of this paper is to prove the existence and uniqueness of the classical solutions of an inverse boundary-value problem. To investigate the solvability of the inverse problem, we carried out a transformation from the original problem to some equivalent auxiliary problem with trivial boundary conditions. Applying the Fourier method and contraction mappings principle, the solvability of the appropriate auxiliary inverse problem is proved. Furthermore, using the equivalency, the existence and uniqueness of the classical solution of the original problem are shown.

scholarly journals Nonlocal inverse boundary-value problem for a 2D parabolic equation with integral overdetermination condition

2020 ◽  
Vol 12 (1) ◽  
pp. 23-33
Author(s):  
E.I. Azizbayov ◽  
Y.T. Mehraliyev
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Original Problem ◽  
Boundary Value ◽  
Auxiliary Problem ◽  
Rectangular Domain ◽  
Unknown Coefficient ◽  
Inverse Boundary ◽  
Inverse Boundary Value Problem

This article studies a nonlocal inverse boundary-value problem for a two-dimensional second-order parabolic equation in a rectangular domain. The purpose of the article is to determine the unknown coefficient and the solution of the considered problem. To investigate the solvability of the inverse problem, we transform the original problem into some auxiliary problem with trivial boundary conditions. Using the contraction mappings principle, existence and uniqueness of the solution of an equivalent problem are proved. Further, using the equivalency, the existence and uniqueness theorem of the classical solution of the original problem is obtained.

scholarly journals Recovery of unknown coefficients in a two-dimensional hyperbolic equation with additional conditions

2021 ◽  
Author(s):  
Elvin Azizbayov ◽  
He Yang ◽  
Yashar Mehraliyev
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Original Problem ◽  
Boundary Value ◽  
Auxiliary Problem ◽  
Fourier Method ◽  
Two Dimensional ◽  
Inverse Boundary ◽  
Inverse Boundary Value Problem

In this paper, a nonlocal inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence (in a certain sense) to the original problem. Then using the Fourier method, solving an equivalent problem is reduced to solving a system of integral equations and by the contraction mappings principle the existence and uniqueness theorem for auxiliary problem is proved. Further, on the basis of the equivalency of these problems the uniquely existence theorem for the classical solution of the considered inverse problem is proved and some considerations on the numerical solution for this inverse problem are presented with the examples.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals On Caputo–Riemann–Liouville Type Fractional Integro-Differential Equations with Multi-Point Sub-Strip Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1899
Author(s):  
Ahmed Alsaedi ◽  
Amjad F. Albideewi ◽  
Sotiris K. Ntouyas ◽  
Bashir Ahmad
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Liouville Type ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results

In this paper, we derive existence and uniqueness results for a nonlinear Caputo–Riemann–Liouville type fractional integro-differential boundary value problem with multi-point sub-strip boundary conditions, via Banach and Krasnosel’skii⏝’s fixed point theorems. Examples are included for the illustration of the obtained results.

scholarly journals Solvability of a fourth order boundary value problem with periodic boundary conditions

1988 ◽  
Vol 11 (2) ◽  
pp. 275-284
Author(s):  
Chaitan P. Gupta
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Elastic Beam ◽  
Periodic Boundary Conditions ◽  
Uniqueness Of Solutions

Fourth order boundary value problems arise in the study of the equilibrium of an elastaic beam under an external load. The author earlier investigated the existence and uniqueness of the solutions of the nonlinear analogues of fourth order boundary value problems that arise in the equilibrium of an elastic beam depending on how the ends of the beam are supported. This paper concerns the existence and uniqueness of solutions of the fourth order boundary value problems with periodic boundary conditions.

scholarly journals Boundary value problem with integral conditions for a linear third-order equation

2003 ◽  
Vol 2003 (11) ◽  
pp. 553-567 ◽  
Author(s):  
M. Denche ◽  
A. Memou
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Strong Solution ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Order Equation ◽  
Integral Conditions ◽  
Third Order ◽  
Integral Boundary

We prove the existence and uniqueness of a strong solution for a linear third-order equation with integral boundary conditions. The proof uses energy inequalities and the density of the range of the generated operator.

Approximate Analytical Solutions to Nonlinear Inverse Boundary Value Problems

2016 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Analytical Solutions ◽  
Traditional Approach ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Inverse Boundary ◽  
Inverse Boundary Value Problem ◽  
Approximate Analytical Solutions ◽  
Inverse Boundary Value Problems

A nontraditional approach to the nonlinear inverse boundary value problem is illustrated using multiple examples of the Poisson equation. The solutions belong to a class of analytical solutions defined through Bézier functions. The solution represents a smooth function of high order over the domain. The same procedure can be applied to both the forward and the inverse problem. The solution is obtained as a local minimum of the residuals of the differential equations over many points in the domain. The Dirichlet and Neumann boundary conditions can be incorporated directly into the function definition. The primary disadvantage of the process is that it generates continuous solution even if continuity and smoothness are not expected for the solution. In this case they will generate an approximate analytical solution to either the forward or the inverse problem. On the other hand, the method does not need transformation or regularization, and is simple to apply. The solution is also good at damping the perturbations in measured data driving the inverse problem. In this paper we show that the method is quite robust for linear and nonlinear inverse boundary value problem. We compare the results with a solution to a nonlinear inverse boundary value problem obtained using a traditional approach. The application involves a mixture of symbolic and numeric computations and uses a standard unconstrained numerical optimizer.

scholarly journals AN INVERSE BOUNDARY VALUE PROBLEM FOR A SECOND ORDER ELLIPTIC EQUATION IN A RECTANGLE

2014 ◽  
Vol 19 (2) ◽  
pp. 241-256 ◽  
Author(s):  
Yashar T. Mehraliyev ◽  
Fatma Kanca
Keyword(s):  
Inverse Problem ◽  
Elliptic Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Second Order ◽  
Order Elliptic Equation ◽  
Inverse Boundary ◽  
Numerical Tests ◽  
Second Order Elliptic Equation

In this paper, the inverse problem of finding a coefficient in a second order elliptic equation is investigated. The conditions for the existence and uniqueness of the classical solution of the problem under consideration are established. Numerical tests using the finite-difference scheme combined with an iteration method is presented and the sensitivity of this scheme with respect to noisy overdetermination data is illustrated.

NON-NEWTONIAN STOKES FLOW WITH FRICTIONAL BOUNDARY CONDITIONS

2007 ◽  
Vol 12 (4) ◽  
pp. 483-495 ◽  
Author(s):  
Fouad Saidi
Keyword(s):  
Fluid Flow ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Variational Inequalities ◽  
Newtonian Fluid ◽  
Stokes Flow ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value

In this work we deal with the boundary value problem for the non‐Newtonian fluid flow with boundary conditions of friction type, mostly by means of variational inequalities. Among others, theorems concerning existence and uniqueness or non‐uniqueness of weak solutions are presented.

A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions

2016 ◽  
Vol 22 (1) ◽  
Author(s):  
Oleg A. Repin ◽  
Svetlana K. Kumykova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Type ◽  
Original Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Fractional Differentiation ◽  
Uniqueness Of Solutions ◽  
Uniqueness Of The Solution ◽  
Equation Of Mixed Type

AbstractThe paper is devoted to the study of a boundary-value problem for an equation of mixed type with generalized operators of fractional differentiation in boundary conditions. We prove uniqueness of solutions under some restrictions on the known functions and on the different orders of the operators of generalized fractional differentiation appearing in the boundary conditions. Existence of solutions is proved by reduction to a Fredholm equation of the second kind, for which solvability follows from the uniqueness of the solution of our original problem.

Sign in / Sign up

Export Citation Format

Share Document