Determination of the unknown source term in a space-fractional diffusion equation

This paper studies an inverse problem of determining the unknown source term in a space-fractional diffusion equation. Three types of spectral regularization method are proposed to deal with the ill-posed problem and the corresponding error estimates are obtained with an a priori strategy to find the regularization parameter. We verify the efficiency of the proposed numerical method with some numerical experiments.     

Multivariate numerical derivative by solving an inverse heat source problem

A method for approximating multivariate numerical derivatives is presented from multidimensional noise data in this paper. Starting from solving a direct heat conduction problem using the multidimensional noise data as an initial condition, we conclude estimations of the partial derivatives by solving an inverse heat source problem with an over-specified condition, which is the difference of the solution to the direct problem and the given noise data. Then, solvability and conditional stability of the proposed method are discussed for multivariate numerical derivatives, and a regularized optimization is adopted for overcoming instability of the inverse heat source problem. For achieving partial derivatives successfully and saving amount of computation, we reduce the multidimensional problem to a one-dimensional case, and give a corresponding algorithm with a posterior strategy for choosing regularization parameters. Finally, numerical examples show that the proposed method is feasible and stable to noise data.     

A numerical method for determining a quasi solution of a backward time-fractional diffusion equation

In this study, we present existence and uniqueness theorems of a quasi solution to backward time-fractional diffusion equation. To do that, we consider a methodology, involving minimization of a least squares cost functional, to identify the unknown initial data. Firstly, we prove the continuous dependence on the initial data for the corresponding forward problem and then we obtain a stability estimate. Based on this, we give the existence theorem of a quasi solution in an appropriate class of admissible initial data. Secondly, it is shown that the cost functional is Fréchet-differentiable and its derivative can be formulated via the solution of an adjoint problem. These results help us to prove the convexity of cost functional and subsequently the uniqueness theorem of the quasi solution. In addition, in order to approximate the quasi solution, WEB-spline finite element method is used. Since the obtained system of linear equations is ill-posed, we apply the Levenberg-Marquardt regularization. Finally, a numerical example is given to show the validation of the introduced method.     

Simultaneous inversion for a diffusion coefficient and a spatially dependent source term in the SFADE

This paper deals with an inverse problem of determining a diffusion coefficient and a spatially dependent source term simultaneously in one-dimensional (1-D) space fractional advection–diffusion equation with final observations using the optimal perturbation regularization algorithm. An implicit finite difference scheme for solving the forward problem is set forth, and a fine estimation to the spectrum radius of the coefficient matrix of the difference scheme is given with which unconditional stability and convergence are proved. The simultaneous inversion problem is transformed to a minimization problem, and existence of solution to the minimum problem is proved by continuity of the input–output mapping. The optimal perturbation algorithm is introduced to solve the inverse problem, and numerical inversions are performed with the source function taking on different forms and the diffusion coefficient taking on different values, respectively. The inversion solutions give good approximations to the exact solutions demonstrating that the optimal perturbation algorithm with the Sigmoid-type regularization parameter is efficient for the simultaneous inversion problem in the space fractional diffusion equation.     

Simultaneous inversion for the diffusion and source coefficients in the multi-term TFDE

This article deals with an inverse problem of determining the space-dependent diffusion coefficient and the source coefficient simultaneously in the multi-term time fractional diffusion equation (TFDE in short) using measurements at one inner point. From a view point of optimality, solving the inverse problem is transformed to minimize an error functional with the help of the solution operator from the unknown to the additional observation. The solution operator is nonlinear but it is of Lipschitz continuity by which existence of a minimum to the error functional is obtained using Sobolev embedding theorems. The homotopy regularization algorithm is introduced to solve the simultaneous inversion problem based on the minimization problem, and numerical examples are presented. The inversion solutions give good approximations to the exact solutions demonstrating that the homotopy regularization algorithm is efficient for the simultaneous inversion problem arising in the multi-term TFDE.     

A 2D problem of time-fractional heat order for two-temperature thermoelasticity under hydrostatic initial stress

The present study solves the problem of thermoelastic interactions in a half-space medium under hydrostatic initial stress in the context of a fractional order heat conduction model with two-temperature theory. The analytical solutions of the field variables are obtained by using the normal mode analysis. The obtained solutions are then applied to a specific problem for a thermally insulated surface which is acted upon by a load. The distributions of the two temperatures, displacements, and the stress components inside the half-space are studied. The graphical results depict that the fractional parameter has significant effects on all the studied field variables. Comparisons are made within the theory in the presence and absence of the hydrostatic initial stress. Thus, we can conclude that the fractional order generalized thermoelasticity model may be an improvement on studying elastic materials.     

Solution of a boundary-value problem for the generalized diffusion equation

The boundary-value problem is solved for the complete diffusion equation and then for the same equation with a chemical reaction taken into account, for the case of a liquid flow for which the width and length are much larger than the thickness.     

Analytical solutions to the planar non-axisymmetric elasticity and thermoelasticity problems for homogeneous and inhomogeneous annular domains

This paper presents an analytical approach to solving the plane non-axisymmetric elasticity and thermoelasticity problems in terms of stresses for isotropic, homogeneous or inhomogeneous annular domains. The key feature of this approach is integration of the equilibrium equations in order to: a) express all the stress-tensor components in terms of a governing stress; b) deduce the integral equilibrium conditions, which are vital for the solution. Because the equilibrium equations are insensitive of material properties, the obtained expressions and integral conditions fit both homogeneous and inhomogeneous cases. The governing stress is derived out of the compatibility equation. Regarding complete construction of the solution, the integral compatibility conditions are deduced by integrating the strain-displacement relations. In the case of inhomogeneous material, the governing compatibility equation is reduced to Volterra type integral equation which then is solved by simple iteration method. The rapid convergence of the iterative procedure is established.     

Determining the initial distribution in space-fractional diffusion by a negative exponential regularization method

In this paper, we consider the backward problem for diffusion equation with space fractional Laplacian, i.e. determining the initial distribution from the final value measurement data. In order to overcome the ill-posedness of the backward problem, we present a so-called negative exponential regularization method to deal with it. Based on the conditional stability estimate and an a posteriori regularization parameter choice rule, the convergence rate estimate are established under a-priori bound assumption for the exact solution. Finally, several numerical examples are proposed to show that the numerical methods are effective.     

Inverse heat transfer analysis for design and control of a micro-heater array

A non-linear inverse heat source identification problem is described and solved. The inverse problem analysis is used in the design of an embedded micro-heater array and to estimate the required control settings, which are the input currents to each heating element, to generate as close as possible to a prescribed temperature profile on the surface of a thin copper film. The purpose of the micro-heater array is to control the local copper microstructure through control of the local temperature field. A finite element model of the micro-heater system is used to define a discrete set of non-linear equations used as a basis for the inverse problem solution. Two methods are explored to solve the inverse problem, a direct minimization method with Tikhonov regularization and a passivity-based feedback control algorithm. A uniform and a linear temperature distribution could be attained in the central region above the micro-heater array, but the temperatures near the edges of the domain could not be controlled due to heat loss at the edges. Thus, to control the temperature field over the full width of the domain, the heater array must extend beyond the domain of interest. Both methods to solve the inverse problem are found to perform well. The regularization method allows for a smoother solution, while the feedback control method is simpler as the coefficient matrix for which the update remains unchanged for each iteration.     

A coupled complex boundary method for the Cauchy problem

Considered in this paper is a Cauchy problem governed by an elliptic partial differential equation. In the Cauchy problem, one wants to recover the unknown Neumann and Dirichlet data on a part of the boundary from the measured Neumann and Dirichlet data, usually contaminated with noise, on the remaining part of the boundary. The Cauchy problem is an inverse problem with severe ill-posedness. In this paper, a coupled complex boundary method (CCBM), originally proposed in [Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Prob. 2014;30:055002], is applied to solve the Cauchy problem stably. With the CCBM, all the data, including the known and unknown ones on the boundary are used in a complex Robin boundary on the whole boundary. As a result, the Cauchy problem is transferred into a complex Robin boundary problem of finding the unknown data such that the imaginary part of the solution equals zero in the domain. Then the Tikhonov regularization is applied to the resulting new formulation. Some theoretical analysis is performed on the CCBM-based Tikhonov regularization framework. Moreover, through the adjoint technique, a simple solver is proposed to compute the regularized solution. The finite-element method is used for the discretization. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.     

On Saint-Venant’s problem for anisotropic, inhomogeneous, cylindrical Cosserat elastic shells

We investigate the static deformation of cylindrical elastic shells, using the theory of Cosserat surfaces. We consider anisotropic and inhomogeneous cylindrical shells with arbitrary (open or closed) cross-section. The constitutive coefficients are assumed to be independent of the axial coordinate. In the context of linearized theory, we determine a solution of the relaxed Saint-Venant’s problem. Finally, we apply the general results in the special cases of circular cylindrical shells and of Cosserat plates made from an orthotropic material.     

A first order method for the Cauchy problem for the Laplace equation using BEM

The purpose is to propose an improved method for inverse boundary value problems. This method is presented on a model problem. It introduces a higher order problem. BEM numerical simulations highlight the efficiency, the improved accuracy, the robustness to noisy data of this new approach, as well as its ability to deblur noisy data.     

Numerical treatment of a skew-derivative problem for the Laplace equation in the exterior of an open arc

The skew-derivative problem for harmonic functions in the exterior of an open arc in a plane is considered. This problem models the electric current in a semiconductor film from an electrode of arbitrary shape in the presence of a magnetic field. A numerical method for solving the problem is proposed. The method is based on a boundary-integral-equation approach. The proposed numerical method is tested. The numerical simulation is presented for different values of the parameters and different shapes of the electrode. Physical effects found in numerical experiments are discussed.