A novel adaptive meshless numerical method for tackling with the two-dimensional time fractional telegraph equations on arbitrary domains based on partitioning domains is presented in this paper. The proposed approach has many advantages. (i) When the exact solution varies dramatically in some parts of the domain and gently in others, our method can reduce the amount of calculation waste and improve computational speed. (ii) With the assistance of the extension theorem, the multiscale orthonormal bases of two-dimensional reproducing kernel space on rectangle regions are constructed to obtain the approximate solutions of the equations on arbitrary domains, which can achieve high accuracy. (iii) The convergence-order analysis of bicubic spline space can be employed to study the convergence order of the proposed strategy primely. Eventually, some numerical examples and comparisons graphically elucidate the implementation and capability of the adaptive meshless method, which is more accurate and efficient than some existing methods.
In this paper, we study the space-time variable-order fractional diffusion equation with a variable diffusion coefficient. The fractional derivatives of variable orders are considered in the Caputo sense. We propose a numerically efficient pseudospectral method with the Chebyshev polynomials as orthogonal basis functions. Also, we examine the error bound and convergence analysis of the approximate solution. A variation in the maximum absolute error with the different variable orders in space and time is studied. Some illustrative examples are presented with different boundary conditions, e.g., Dirichlet, mixed, and non-local. The method's applicability is also tested with the problem of owning fractional power in the solution. The results obtained from the proposed approach demonstrate the efficacy and reliability of the method.
The Riesz fractional derivative has been employed to describe the spatial derivative in a variety of mathematical models. In this work, the accuracy of the finite element method (FEM) approximations to Riesz fractional derivative was enhanced by using adaptive refinement. This was accomplished by deducing the Riesz derivatives of the FEM bases to work on non-uniform meshes. We utilized these derivatives to recover the gradient in a space fractional partial integro-differential equation in the Riesz sense. The recovered gradient was used as an a posteriori error estimator to control the adaptive refinement scheme. The stability and the error estimate for the proposed scheme are introduced. The results of some numerical examples that we carried out illustrate the improvement in the performance of the adaptive technique.
In this paper, we continue the study of viscous Burgers’ equations by Il’in and Oleinik for single model equation with convex nonlinearity [6] by introducing the nonlinear degenerate viscosity. Since the degeneracy of this paper is considered, then there are two conditions for estimate of the traveling fronts U when m≥ 1 and 00 and m>b . Moreover, the following Taylor expansion is employed f(U + π _z ) - f^' (U)π _z - f(U) = ∫_0^1( (1 - s)f^ (U + sπ _z )ds)π _z^2 = 𝒪(1)π _z^2 to overcome the estimate of term F_1 , where this term is transformation result of nonlinearity for first order derivative in (1). The stability of traveling fronts U is presented to give the information how close the distance between the solution u of (1) and the traveling fronts U is under the small perturbations. This stability result is based on the energy estimates under the condition N(t)≤ Dm(u_++u_-) . To validate our works and to illustrate the effect of nonlinear degenerate viscosity, the numerical simulations are provided by using the standard finite difference for discretization steps.
The restarted global Krylov subspace methods are popular to solve matrix equations. Although these methods reduce storage costs, some important information is lost at the time of restart and this slows down the convergence. However, it is possible to keep some crucial information from the previous cycle and then apply them at the restart. To retain this information, it is necessary to update the starting block vector. For this purpose, we introduce the implicitly restarted global Arnoldi process that is based on the implicit double-shift QR iteration. Moreover, we develop the implicitly restarted global FOM and GMRES methods to speed up the convergence. In the mentioned methods, a starting block vector is selected so that the smallest eigenvalues in magnitude are deflated, and the corresponding approximate block Ritz vectors and harmonic Ritz vectors are augmented to the current Krylov subspace, respectively. It is demonstrated that the deflation of the smallest eigenvalues can be particularly imperative for global methods. Finally, the efficiency of these methods are appraised on two academic examples as well as on a case study in surface fitting application.
