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In this study, we examine a modified heat equation with memory and nonlinear source. The source function is considered under two different conditions, the global Lipschitz and the exponential growth functions. For the first condition, a special weighted Banach space is applied to deduce a desired result without any assumption on sufficiently small time and initial data. For the second condition of exponential growth, we apply the Moser–Trudinger inequality to cope with the source function, and a special time-space norm to deduce the unique existence of a global solution in regard to sufficiently small data. The main objective of this work is to prove the global existence and uniqueness of mild solutions. Besides the solution techniques, our main arguments are also based on the Banach fixed point theorem and linear estimates for the mild solution. The highlight of this study is that it is the first work on the global well-posedness for the mild solution of the fractional heat conduction with memory and nonlinear sources. [ABSTRACT FROM AUTHOR]

In this work, we focus on the Cauchy problem for the Poisson equation in the two dimensional domain, where the initial data is disturbed by random noise. In general, the problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. To regularize the instable solution of the problem, we have applied a nonparametric regression associated with the truncation method. Eventually, a numerical example has been carried out, the result shows that our regularization method is converged; and the error has been enhanced once the number of observation points is increased. [ABSTRACT FROM AUTHOR]

In this paper, we study on an inverse problem to determine an unknown source term in a time fractional diffusion equation, whereby the data are obtained at the later time. In general, this problem is illposed, therefore the Tikhonov regularization method is proposed to solve the problem. In the theoretical results, a priori error estimate between the exact solution and its regularized solutions is obtained. We also propose two methods, a priori and a posteriori parameter choice rules, to estimate the convergence rate of the regularized methods. In addition, the proposed regularized methods have been verified by numerical experiments to estimate the errors between the regularized solutions and exact solutions. Eventually, from the numerical results it shows that the posteriori parameter choice rule method converges to the exact solution faster than the priori parameter choice rule method. [ABSTRACT FROM AUTHOR]

In this paper, we solve the Cauchy problem for an inhomogeneous Helmholtz-type equation with homogeneous Dirichlet and Neumann boundary condition. The proposed problem is ill-posed. Up to now, most investigations on this topic focus on very specific cases, and with Dirichlet boundary condition. Recently, we solve this problem in 2D for an inhomogeneous modified Helmholtz equation (2012). This work is a continuous expansion of our previous results. Herein we introduce a general filter regularization (GFR) method, and then from the GFR we deduce two concrete filters, which are a foundation to implement a numerical procedure. In addition, we develop a numerical model for solving this problem in three dimensional region. The proposed filter method has been verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are going to consider the following problemwith corresponding measured data functions , a given function f is known as the “source function”. We want to determine the solution for from the Cauchy data at . The problem is ill-posed, as the solution exhibits unstable dependence on the given data functions. The aim of this paper is to introduce new efficient regularization methods, such as truncation of high frequency and quasi-boundary-type methods, with explicit error estimates for an extended case (i.e. the inhomogeneous problem with in Eq. (1)). In addition, we also carry out numerical experiments and compare numerical results of our methods with Qin and Wei’s methods [1] in homogeneous case, as well as to compare our numerical solutions with exact solutions in inhomogeneous case. It shows that our truncation and Qin and Wei’s truncation methods giving almost same results. However, our quasi-boundary-type methods show a better results than quasi-reversibility method of Qin and Wei in term of error estimation and convergence speed. [Copyright &y& Elsevier]

This work is concerned with solving the Cauchy problem for quasilinear elliptic equations whose exponential instability is manifestly seen by the catastrophic growth in the representation of the exact solution. Our proposed regularization procedure consists in damping the unbounded terms in the representation. Moreover, we show that its solution converges to the exact solution uniformly and strongly in L 2 under a priori assumptions on the exact solution. In order to verify our analysis and the accuracy of the numerical procedures, we exhibit two numerical examples. Our main tools for simulation are the trigonometric polynomial approximation, and the fast Fourier transform in combination with the cubic Hermite interpolation. [ABSTRACT FROM AUTHOR]