Mathematical Researches

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This manuscript deals with a Shannon wavelet regularization method to solve the inverse Cauchy problem associated with the Helmholtz equation which uses to identify the radiation wave of an infinite “strip” domain. In view of Hadamard, the proposed problem extremely suffers from an intrinsic ill-posedness, i.e., the exact solution of this problem is computationally impossible to measure since any measurement or numerical computation is polluted by inevitable errors. To retrieve the solution, a regularization scheme based on Shannon wavelet is developed. The regularized solution is restored by Shannon wavelet projection on elements of Shannon multiresolution analysis. Furthermore, the concepts of convergence rate and stability of the proposed scheme are investigated and some new optimal stable estimates of the so-called Holder-Logarithmic type are rigorously derived by imposing an a priori information controlled by Sobolev scale.  The computational performance of the proposed method effectively confirms the applicability and validity of our qualitative analysis../files/site1/files/72/15Abstract.pdf

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In this paper, a high-order numerical method is designed and implemented to solve a boundary value problem governed by the variable-order fractional diffusion equation. This equation contains a variable-order fractional time-derivative and a second-order spatial-derivative. To develop this novel method, a compact finite difference formula and a weighted shifted Grunwald-Letnikov operator are used for spatial and temporal discretization, respectively. It is shown that this method is of fourth- and second-order of convergence accuracy in spatial and time directions, respectively. Also, the solvability, stability and convergence of the peresent method are investigated. To verify the efficiency and high accuracy of this method, some numerical examples and comparative results are presented.