Numerical Algorithms最新文献

In this contribution we deal with Eta functions and their representations as fractional derivatives and fractional integrals. A class of fractional Sturm-Liouville eigenvalue problems is studied. The analytic representation of their eigensolutions is pointed out as well as the orthogonality of the corresponding eigenfunctions.

In this paper we describe the efficient numerical implementation of Fractional HBVMs, a class of methods recently introduced for solving systems of fractional differential equations. The reported arguments are implemented in the Matlab(^{copyright } ) code fhbvm, which is made available on the web. An extensive experimentation of the code is reported, to give evidence of its effectiveness.

In this paper, we develop a novel higher-order compact finite difference scheme for solving systems of Lane-Emden-Fowler type equations. Our method can handle these problems without needing to remove or modify the singularity. To construct the method, initially, we create a uniform mesh within the solution domain and develop a new efficient compact difference scheme. The presented method approximates the derivatives at the boundary nodal points to effectively handle the singularity. Using a matrix analysis approach, we discuss theoretical issues such as consistency, stability, and convergence. The theoretical order of the method is consistent with the numerical convergence rates. To showcase the method’s effectiveness, we apply it to solve various real-life problems from the literature and compare its performance with existing methods. The proposed method provides better numerical approximations than existing methods and offers high-order accuracy using fewer grid points.

In this paper, we propose a novel fully implicit numerical scheme that satisfies both nonlinear energy stability and maximum principle for the space fractional Allen-Cahn equation. Especially, the fully implicit second-order scheme in time has never been proved to preserve the maximum principle before. For the resulting nonlinear scheme, we also propose a nonlinear iterative algorithm, which is uniquely solvable, convergent, and can preserve discrete maximum principle in each iterative step. Then we provide an error estimate by using the established maximum principle which plays a key role in the analysis. Several numerical experiments are presented to verify the theoretical results.

The present article aims to design and analyze efficient first-order strong schemes for a generalized Aït-Sahalia type model arising in mathematical finance and evolving in a positive domain ((0, infty )), which possesses a diffusion term with superlinear growth and a highly nonlinear drift that blows up at the origin. Such a complicated structure of the model unavoidably causes essential difficulties in the construction and convergence analysis of time discretizations. By incorporating implicitness in the term (alpha _{-1} x^{-1}) and a corrective mapping (Phi _h) in the recursion, we develop a novel class of explicit and unconditionally positivity-preserving (i.e., for any step-size (h>0)) Milstein-type schemes for the underlying model. In both non-critical and general critical cases, we introduce a novel approach to analyze mean-square error bounds of the novel schemes, without relying on a priori high-order moment bounds of the numerical approximations. The expected order-one mean-square convergence is attained for the proposed scheme. The above theoretical guarantee can be used to justify the optimal complexity of the Multilevel Monte Carlo method. Numerical experiments are finally provided to verify the theoretical findings.

This article provides an effective numerical approach using the fractional integral operational matrix method for a fractional Legendre wavelet to deal with multi-dimensional fractional optimal control problems. We proposed operational matrices and implemented them to simplify multi-dimensional fractional optimal control problems into a set of equations, utilizing well-known formulas such as the Caputo-Fabrizio operator with a non-singular kernel defined for calculating fractional derivatives and integrals of fractional Legendre wavelets. Finally, the Lagrange multiplier technique is applied, and we get the state and control functions. The convergence analysis and error bounds of the proposed scheme are established. To check the veracity of the presented method, we tested numerical examples using the fractional Legendre wavelet method and obtained the cost function value based on identifying state and control functions.

This study presents a novel high-order numerical method designed for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The Caputo definition is employed to characterize the time-fractional derivative. A weak singularity at the initial time ((t=0)) is encountered in the considered problem. To overcome this, we consider the high-order L2-1(_sigma ) formula on a suitably designed non-uniform fitted mesh, to discretize the time-fractional derivative. Further, a high-order two-dimensional compact operator is developed to approximate the spatial variables. Moreover, an alternating direction implicit (ADI) approach is designed to solve the resulting system of equations by decomposing the two-dimensional problem into two separate one-dimensional problems. The theoretical analysis, encompassing both stability and convergence aspects, is conducted comprehensively. More precisely, it is shown that method is convergent of order (mathcal Oleft( {N_t^{-min {3-alpha ,theta alpha ,1+alpha ,2}}}+h_x^4+h_y^4right) ), where (alpha in (0,1)) represents the order of the fractional derivative, (theta ) is a parameter which is utilized in the construction of the fitted mesh, (N_t) is the temporal discretization parameter, and (h_x) and (h_y) represent the spatial mesh widths. The numerical outcomes for three test problems, each featuring the nonsmooth solution, verified the theoretical findings. Further, the proposed method on fitted meshes exhibits superior numerical accuracy in comparison to the existing methods.

In this paper, we construct numerical methods of fourth, sixth and eighth orders convergence for solving fully fourth order nonlinear differential equation with the Dirichlet boundary conditions. The methods are based on the use of the trapezoidal quadrature formula with corrections for computing integrals at each iteration of the continuous iterative method for finding the solutions of the BVP. We get the error estimates for the actually obtained numerical solutions of the problem. Many numerical examples confirm the theoretical conclusions and show the efficiency of the proposed methods in comparison with some existing methods.

Based on the diagonal and off-diagonal splitting (DOS) iteration scheme (Dehghan et al. Filomat 31(5), 1441–1452 2017), we offer an iteration procedure called Uzawa-DOS to solve a class of saddle-point problems (SPPs). Each iteration of this iterative method involves two subsystems with diagonal and lower triangular matrices. Due to the simple structure of involved coefficient matrices, two linear subsystems are solvable exactly, which is a notable precedence of the Uzawa-DOS method and can make it inexpensive to execute. Theoretical analysis verifies convergence of the proposed method under appropriate conditions. The suggested method is validated by numerical experiments.

In this paper, by regarding the two-subspace Kaczmarz method as an alternated inertial randomized Kaczmarz algorithm we present a better convergence rate estimate under a mild condition. Furthermore, we accelerate the alternated inertial randomized Kaczmarz algorithm and introduce a multi-step inertial randomized Kaczmarz algorithm which is proved to have a faster convergence rate. Numerical experiments support the theory results and illustrate that the multi-inertial randomized Kaczmarz algorithm significantly outperform the two-subspace Kaczmarz method in solving coherent linear systems.