Optimization method for the inverse problem of reconstructing the source term in a parabolic equation

This work investigates the inverse problem of reconstructing a spacewise dependent heat source in the parabolic heat equation using a final temperature measurement. Such problem has important application in a large field of applied science. On the basis of the optimal control framework, the existence and necessary condition of the minimizer for the cost functional are established. The global uniqueness and stability of the minimizer are deduced from the necessary condition. The Landweber iteration algorithm is applied to the inverse problem and some numerical results are presented for various typical test examples.     

Numerical analysis and physical simulations for the time fractional radial diffusion equation

We do the numerical analysis and simulations for the time fractional radial diffusion equation used to describe the anomalous subdiffusive transport processes on the symmetric diffusive field. Based on rewriting the equation in a new form, we first present two kinds of implicit finite difference schemes for numerically solving the equation. Then we strictly establish the stability and convergence results. We prove that the two schemes are both unconditionally stable and second order convergent with respect to the maximum norm. Some numerical results are presented to confirm the rates of convergence and the robustness of the numerical schemes. Finally, we do the physical simulations. Some interesting physical phenomena are revealed; we verify that the long time asymptotic survival probability ∝t^−α, but independent of the dimension, where α is the anomalous diffusion exponent.     

A monotone Newton multisplitting method for the nonlinear complementarity problem with a nonlinear source term

In this paper, we develop a Newton multisplitting method for the nonlinear complementarity problem with a nonlinear source term in which the multisplitting method is used as secondary iterations to approximate the solutions for the resulting linearized subproblems. We prove the monotone convergence theorem for the proposed method under proper conditions.     

A tau approach for solution of the space fractional diffusion equation

Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use.     

Numerical procedure for the determination of an unknown parameter in a parabolic equation

A numerical procedure for an inverse problem of determining unknown source parameter of one-dimensional parabolic equation subject to the specification of the solution at internal point along with the usual initial boundary conditions is considered. By using some transformation the problem is reformulated to a nonlocal parabolic problem. Some numerical examples using the proposed numerical procedure are presented.     

A new approach of superconvergence analysis for two-dimensional time fractional diffusion equation

In this paper, a new approach of superconvergent estimate of bilinear finite element is established for two-dimensional time-fractional diffusion equation under fully-discrete scheme. The novelty of this approach is the combination technique of the interpolation and Ritz projection as well as the superclose estimate in $H^1$-norm between them, which avoids the difficulty of constructing a postprocessing operator for Ritz projection operator, and reduces the regularity requirement of the exact solution. At the same time, three numerical examples are carried out to verify the theoretical analysis.     

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACT

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.     

Uniqueness and numerical scheme for the Robin coefficient identification of the time-fractional diffusion equation

We study an inverse problem of determining the Robin coefficient of fractional diffusion equation from a nonlocal boundary condition. Based on the property of Caputo fractional derivative, the uniqueness is proved. The numerical schemes for the direct problem and the inverse problem are developed. Three examples are given to show the effectiveness of the presented methods.     

Optimal control of a fractional diffusion equation with state constraints

This paper is concerned with the state constrained optimal control problems of a fractional diffusion equation in a bounded domain. The fractional time derivative is considered in the Riemann-Liouville sense. Under a Slater type condition we prove the existence a Lagrange multiplier and a decoupled optimality system.     

Numerical solution for a class of space-time fractional equation by the piecewise reproducing kernel method

ABSTRACT

Due to the non-locality of fractional derivative, the analytical solution and good approximate solution of fractional partial differential equations are usually difficult to get. Reproducing kernel space is a perfect space in studying this type of equations, however the numerical results of equations by using the traditional reproducing kernel method (RKM) isn't very good. Based on this problem, we present the piecewise technique in the reproducing kernel space to solve this type of equations. The focus of this paper is to verify the stability and high accuracy of the present method by comparing the absolute error with traditional RKM and study the effect on absolute error for different values of α. Furthermore, we can study the distribution of entire space at a particular time period. Three numerical experiments are provided to verify the efficiency and stability of the proposed method. Meanwhile, it is tested by experiments that the change of the value of α has little effect on its accuracy.     

An asymptotic numerical method for fourth order singular perturbation problems with a discontinuous source term

Singularly perturbed two-point boundary-value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter ? multiplying the highest derivative, and suitable boundary conditions. In this paper a computational method for solving this system is presented. In this method we first find the zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the numerical method, which is constructed for this problem and which involves an appropriate piecewise-uniform mesh.     

Existence of a solution for the fractional differential equation with nonlinear boundary conditions

Using the method of upper and lower solutions and its associated monotone iterative, we present an existence theorem for a nonlinear fractional differential equation with nonlinear boundary conditions.     

Identification of variable spacial coefficients for a beam equation from boundary measurements

We consider a system described by the Euler-Bernoulli beam equation, with one end clamped and with torque input at the other end. The output function are the displacement and the angle velocity at the non-clamped end of the beam. We study the identification of the spatially variable coefficients in the beam equation, from input-output data. We show that both the density and the flexural rigidity of the beam (which are assumed to be of class C⁴) can be uniquely determined if the input and output functions are known for all positive times.     

The use of entropy minimization for the solution of blind source separation problems in image analysis

Entropy minimization is closely associated with pattern recognition. The present contribution uses a direct minimization of an entropy like function to solve the blind source separation problem for image reconstruction. The mixture patterns are decomposed using SVD and then global stochastic optimization is used to find the first irreducible image pattern. Further images are then subsequently reconstructed, by imposing a 2D correlation coefficient for dissimilarity to prevent repeated images, until all images are exhaustively enumerated. Three test cases are used, including (1) a set of three black and white texturally different photographs (2) a set of three RGB geometrically similar photographs and (3) an underdetermined problem involving an imbedded watermark. Cases 1 and 2 are easily solved with outstanding image quality. Both searches are conducted in an unsupervised manner—no a priori information is used. In Case 3, the watermark is enhanced after targeting the region for entropy minimization. The present results have a wide variety of applications, including image and spectroscopic analysis.     

Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem

In this paper, we study hybrid fuzzy differential equation initial value problems (IVPs). We consider the problem of finding their numerical solutions by using a recent characterization theorem of Bede for fuzzy differential equations. We prove a corollary to Bede’s characterization theorem and give a characterization theorem for hybrid fuzzy differential equation IVPs. Then we prove that any suitable numerical method for ODEs can be applied piecewise to numerically solve hybrid fuzzy differential equation IVPs. Numerical examples are provided which connect the new results with previous findings.     

Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method

In this paper, we consider the identification of a corrosion boundary for the two-dimensional Laplace equation. A boundary collocation method is proposed for determining the unknown portion of the boundary from the Cauchy data on a part of the boundary. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization technique, while the regularization parameter is provided by the generalized cross-validation criterion. Numerical examples show that the proposed method is reasonable and feasible.     

An approach for the identification of diffusion coefficients in the quasi-steady state of a post-discharge nitriding process

This work presents a model for the nitrogen concentration in iron nitride layers at the quasi-steady state and a method for computing the diffusion coefficients for post-discharge nitriding. The method is based upon an inverse problem of coefficient identification. The related moving boundary diffusion problem is modeled assuming constant diffusion coefficients and taking into account the observed qualitative behavior of the post-discharge nitriding process. To develop an algorithm for the identification of the diffusion coefficients the solution of the least square problem is transformed, with some algebraic manipulations, into a simple geometrical rule.     

Efficient difference scheme for numerical solution of a convective diffusion problem

Problems of modeling of atmospheric circulation are investigated. A new method for solution of a one-dimensional nonstationary inhomogeneous initial-boundary-value problem of convective diffusion is considered. The problem is solved using a new unconditionally stable and efficient difference scheme. The results of a theoretical analysis of the scheme are presented. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 64–74, May–June 2007. An erratum to this article is available at .     

Monotone methods for a finite difference system of reaction diffusion equation with time delay

The aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction-diffusion equations with time delays. The iterative schemes lead to computational algorithms as well as existence, uniqueness, and upper and lower bounds of the solution. An application to a diffusive logistic equation with time delay is given.