Salih Tatar

Alfaisal University · Department of Mathematics & Computer Science

In this paper, we study simultaneous determination of the strain hardening exponent, the shear modulus and the yield stress in an inverse problem. First, we analyze the direct and the inverse problems. Then we formulate the inverse problem in the Bayesian framework. After solving the direct problem by an iterative approach, we propose a numerical m...

In this paper, we study an inverse problem for identifying the initial value in a space-time fractional diffusion equation from the final time data. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. It is proved that the inverse problem is an ill-posed proble...

This paper is devoted to an inverse problem for a nonlinear parabolic equation related to brain tumor dynamics. After reformulating the inverse problem as a minimization problem, we prove the existence and stability of the solution to the minimization problem. Based on the Fréchet differentiability of the objective (cost) functional, we develop an...

In this paper, we study the flux identification problem for a nonlinear time-fractional viscoelastic equation with a general source function based on the boundary measurements. We prove that the direct problem is well-posed, i.e., the solution exists, unique and depends continuously on the heat flux. Then the Fr\'echet differentiability of the cost...

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has a quasi-solution. The direct problem is solved by the method of lines using an operato...

In this paper, we develop an iterative method of lines scheme for the numerical solution to the time fractional Richards equation with implicit Neumann boundary conditions, which is an effective tool for describing a process of flow through unsaturated media. A numerical example is provided to show the effectiveness of the presented method for diff...

Direct and inverse problems for a nonlinear time fractional equation are studied. It is proved that the direct problem has a unique weak solution and the solution depends continuously on the nonlinear coefficient. Then it is shown that the inverse problem has a quasi-solution. The direct problem is solved by method of lines using an operator approa...

In this paper, we consider an inverse source problem for a fractional pseudo-parabolic equation. We show that the problem is severely ill-posed (in the sense of Hadamard) and the Tikhonov regularization method is proposed to solve the problem. In addition, we present numerical examples to illustrate applicability and accuracy of the proposed method...

In this paper, we consider recovery of solute concentration and dispersion flux in an inhomogeneous time fractional diffusion equation. We prove that the considered problem is ill-posed, i.e. the solution does not depend continuously on the data. In order to obtain a regularized solution, we propose a truncation regularization method. The convergen...

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded...

This paper is devoted to numerical solutions of direct and inverse problems for the nonlinear nonlocal time fractional equation ∂β/∂tβu(x,y,t)-∇·[f(T²)∇u]=2t, where T²=|∇u|². After solving the direct problem by the method of lines, a numerical method based on discretization of the minimization problem, steepest descent method, and least-squares app...

In this paper, we study an inverse problem for an inhomogeneous time-fractional diffusion equation in the one-dimensional semi-infinite domain. Such a problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative. After we show that the inverse problem is severely ill-pose...

In this paper, we consider recovery of solute concentration and dispersion flux in an inhomogeneous time fractional diffusion equation. We prove that the considered problem is ill-posed, i.e. the solution does not depend continuously on the data. In order to obtain a regularized solution, we propose a truncation regularization method. The convergen...

This paper is devoted to simultaneous determination of the strain hardening exponent, the shear modulus and the elastic stress limit in an inverse problem. The inverse problem consists of determining the unknown coefficient in the nonlinear equation by measured output data (or additional data) given in the integral form. After we solve direct probl...

A nonlinear time-fractional inverse coefficient problem is considered. The unknown coefficient depends on the solution. It is proved that the direct problem has a unique solution. Afterwards the continuous dependence of the solution of the corresponding direct problem on the coefficient is proved. Then the existence of a quasi-solution of the inver...

This paper is concerned with the biological neuron model of Morris-Lecar system of equations (Morris and Lecar, 1981) [17]. We prove the existence and uniqueness of strong solutions. In addition, we prove the continuous dependence of solutions to the leak conductance parameter.

In this article, we consider the problem of identifying an unknown coefficient in a nonlinear diffusion equation. Under appropriate conditions, we prove the existence and the uniqueness of solution for the inverse problem. For the numerical solution of the inverse problem, a numerical method based on discretization of the minimization problem, stee...

This study is devoted to a nonlinear time fractional inverse coecient problem. The unknown coecient depends on the gradient of the solution and belongs to a set of admissible coecients. First we prove that the direct problem has a unique solution. Afterwards we show the continuous dependence of the solution of the corresponding direct problem on th...

Fractional(nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues and they are used to model anomalous diffusion, especially in physics. This paper deals with a nonlocal inverse source problem for a one dimensional space-time fractional diffusion equation ∂ β t u = −r β(−∆)α/2u(t, x...

This paper is devoted to the determination of an unknown function that describes elastoplastic properties of a bar under torsion. The mathematical (evolution) model leads to an inverse problem that consists of determining the unknown coefficient (Formula presented.), in the nonlinear parabolic equation (Formula presented.), using measured output da...

Fractional(nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues and they are used to model anomalous diffusion, especially in physics. This paper deals with a nonlocal inverse source problem for a one-dimensional space-time fractional diffusion equation where and . At first we def...

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues and they are used to model anomalous diffusion, especially in physics. This paper is devoted to a nonlocal inverse problem related to the space-time fractional equation . The existence of the solution for the inverse...

In this study, an effective modification of the semi-analytic inversion method is presented. The semi-analytic inversion method is developed to solve an inverse coefficient problem arising in materials science instead of the parametrization method as a different and stronger method. The inverse coefficient problem is related to reconstruction of th...

This study is devoted to the numerical solution of an inverse coefficient problem for a density dependent nonlinear reaction-diffusion equation. The method is based on approximating the unknown coefficient by polynomials. An optimal idea for solving the inverse problem is to minimize an error functional between the output data and the additional da...

In this paper, monotonicity of input-output mapping related to inverse elastoplastic torsional problem is studied. In this context, torsional behavior of power-hardening engineering materials is investigated. The equation of the elastoplastic torsion of a strain hardening bar is given by Au := -del . (g vertical bar del u vertical bar(2)del u). Alt...

This paper analyzes the existence and the uniqueness problem for an n‐dimensional nonlinear inverse reaction‐diffusion problem with a nonlinear source. A transformation is used to obtain a new inverse coefficient problem. Then, a parabolic differential operator L λ is defined to establish the relation between the solution of L λ = 0 and the new inv...

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by fractional-order derivatives. This article considers a nonlocal inverse problem and shows that the exponents of the fractional time and space derivatives are determined uniquely by the data $u(t, 0)= g(t),\; 0 < t < T$. The uniqueness result is a th...

This paper is related to a class of inverse problems for identification of the coefficient in a square porous medium. The unknown coefficient depends on the solution and belongs the class of admissible coefficients. Using continuous dependence of solutions on the coefficient convergence in the corresponding direct problems, the existence of the qua...

a b s t r a c t This paper is devoted to some class of inverse coefficient problems. By using a well-known transformation, the inverse problem is transformed to a new problem without the unknown time dependent coefficient. Therefore, the new inverse problem can be solved easily. To show the efficiency of the present method, some examples are presen...

This paper is devoted to the determination of an unknown function that describes elastoplastic properties of a bar under torsion. The mathematical (evolution) model leads to an inverse problem that consists of determining the unknown coefficient g = g(xi(2)), xi(2) = vertical bar del u vertical bar(2), in the nonlinear parabolic equation u(t) - del...

A new method of determining elastoplastic properties of a beam from an experimentally given value T, T (φ) of torque (or torsional rigidity), during the quasistatic process of torsion, given by the angle of twist φ ∈ [φ*, φ *], is proposed. The mathematical model leads to the inverse problem of determining the unknown coefficient g = g (ξ2), ξ = |...

An inverse problem related to the determination of elastoplastic properties of a beam is considered within J 2 deformation theory. A new fast algorithm is proposed for the identification of elastoplastic properties of engineering materials. This algorithm is based on finding the three main parameters of the unknown curve g(ξ²) (plasticity function)...

Linear and nonlinear boundary value problems related to elastic and elastoplastic torsional rigidity of a beam are considered. As a sample model, pure elastic torsion problem for a square cross section bar is solved by the method of separation variables. The well-known analytical formula tot torsional rigidity is obtained, The nonlinear boundary va...