Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics

This paper deals with approximating the time fractional Tricomi-type model in the sense of the Caputo derivative. The model is often adopted for describing the anomalous process of nearly sonic speed gas dynamics. The temporal semi-discretization is computed via a finite difference algorithm, while the spatial discretization is obtained using the local radial basis function in a finite difference mode. The local collocation method approximates the differential operators using a weighted sum of the function values over a local collection of nodes (named stencil) through a radial basis function expansion. This technique considers merely the discretization nodes of each subdomain around the collocation node. This leads to sparse systems and tackles the ill-conditioning produced of global collocation. The theoretical convergence and stability analyses of the proposed time semi-discrete scheme are proved by means of the discrete energy method. Numerical results confirm the accuracy and efficiency of the new approach.The authors are very grateful to the reviewers for their valuable comments on the manuscript that led to many improvements.info:eu-repo/semantics/publishedVersio

Chebyshev cardinal functions for solving volterra-fredholm integro- differential equations using operational matrices

Abstract In this paper, an effective direct method to determine the numerical solution of linear and nonlinear Fredholm and Volterra integral and integro-differential equations is proposed. The method is based on expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are described in detail. These matrices play the important role of reducing an integral equation to a system of algebraic equations. Illustrative examples are shown, which confirms the validity and applicability of the presented technique

A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer

Introduction: During the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arises in porous media. The modified time-fractional diffusion equation provides a deeper understanding of several dynamic phenomena. Objectives: The purpose of the paper is to develop an efficient meshless technique for approximating the modified time-fractional diffusion problem formulated in the Riemann–Liouville sense. Methods: The temporal discretization is performed by integrating both sides of the modified timefractional diffusion model. The unconditional stability of the time discretization scheme and the optimal convergence rate are obtained. Then, the spatial derivatives are discretized through a local hybridization of the cubic and Gaussian radial basis function. This hybrid kernel improves the condition of the system matrix. Therefore, the solution of the linear system can be obtained using direct solvers that reduce significantly computational cost. The main idea of the method is to consider the distribution of data points over the local support domain where the number of points is almost constant. Results: Three examples show that the numerical procedure has good accuracy and applicable over complex domains with various node distributions. Numerical results on regular and irregular domains illustrate the accuracy, efficiency and validity of the technique.The authors would like to thank the editors and three anonymous reviewers for their insightful comments and suggestions that greatly improved the quality of this paper.info:eu-repo/semantics/publishedVersio

An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model

This paper adopts an efficient meshless approach for approximating the nonlinear fractional fourth-order diffusion model described in the Riemann–Liouville sense. A second-order difference technique is applied to discretize temporal derivatives, while the radial basis function meshless generated the finite difference scheme approximates the spatial derivatives. One key advantage of the local collocation method is the approximation of the derivatives via the finite difference formulation, for each local-support domain, by deriving the basis functions expansion. Another advantage of this method is that it can be applied in problems with non-regular geometrical domains. For the proposed time discretization, the unconditional stability is examined and an error bound is obtained. Numerical results illustrate the applicability and validity of the scheme and confirm the theoretical formulation.The authors are very grateful to the anonymous referees and the editor for useful comments that led to a great improvement of the paper.info:eu-repo/semantics/publishedVersio

Numerical study of the nonlinear anomalous reaction–subdiffusion process arising in the electroanalytical chemistry

This paper presents a meshless method based on the finite difference scheme derived from the local radial basis function (RBF-FD). The algorithm is used for finding the approximate solution of nonlinear anomalous reaction–diffusion models. The time discretization procedure is carried out by means of a weighted discrete scheme covering second-order approximation, while the spatial discretization is accomplished using the RBF-FD. The theoretical discussion validates the stability and convergence of the time-discretized formulation which are analyzed in the perspective to the H1-norm. This approach benefits from a local collocation technique to estimate the differential operators using the weighted differences over local collection nodes through the RBF expansion. Two test problems illustrate the computational efficiency of the approach. Numerical simulations highlight the performance of the method that provides accurate solutions on complex domains with any distribution node type.The authors express their deep gratitude to the referees and the editor for their valuable comments and suggestions.info:eu-repo/semantics/publishedVersio

Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport

This paper focusses on the numerical solution of the nonlinear time-fractional telegraph equation formulated in the Caputo sense. This model is a useful description of the neutron transport process inside the core of a nuclear reactor. The proposed method approximates the unknown solution with the help of two main stages. At a first stage, a semi-discrete algorithm is obtained by means of a difference approach with the accuracy O(τ3−β), where 1<β<2 is the fractional-order derivative. At a second stage, a full-discretization is obtained by an efficient augmented local radial basis function finite difference (LRBF-FD). This method approximates the derivatives of an unknown function at a given point named as center, based on the finite difference at each local-support domain, instead of applying the entire set of points. The technique produces a sparse matrix system, reduces the computational effort and avoids the ill-conditioning derived from the global collocation. The unconditional stability and convergence of the time-discretized formulation are demonstrated and confirmed numerically. The numerical results highlight the accuracy and the validity of the method.info:eu-repo/semantics/publishedVersio

Numerical simulation of a degenerate parabolic problem occurring in the spatial diffusion of biological population

This paper studies a localized meshless algorithm for calculating the solution of a nonlinear biological population model (NBPM). This model describes the dynamics in the biological population and may provide valuable predictions under different scenarios. The solution of the NBPM is approximated by means of local radial basis function based on the partition of unity (LRBF-PU) technique. First, the partial differential equation (PDE) is converted into a system of ordinary differential equations (ODEs) with the help of radial kernels. Afterwards, the system of ODEs is solved through an ODE solver of high-order. The major advantage of this scheme is that it does not requires any linearization. The LRBF-PU approximation helps handling the issue of the matrix ill conditioning that arises in a global RBF approximation. Three examples highlight the efficiency and accuracy of the numerical method. It is verified that the proposed strategy is more efficient in terms of computational time and accuracy in comparison with others available in the literature.The authors would like to express our sincere thanks to the editors and anonymous reviewers who have generously given up valuable time to review which helps to improve the quality of this work.info:eu-repo/semantics/publishedVersio

Numerical approach for modeling fractional heat conduction in porous medium with the generalized Cattaneo model

The generalized Cattaneo model describes the heat conduction system in the perspective of time-nonlocality. This paper proposes an accurate and robust meshless technique for approximating the solution of the time fractional Cattaneo model applied to the heat flow in a porous medium. The fractional derivative is formulated in the Caputo sense with order 1<α<2 . First, a finite difference technique of convergence order O(δt3−α) is adopted to achieve the temporal discretization. The unconditional stability of the method and its convergence are analysed using the discrete energy technique. Then, a local meshless method based on the radial basis function partition of unity collocation is employed to obtain a full discrete algorithm. The matrices produced using this localized scheme are sparse and, therefore, they are not subject to ill-conditioning and do not pose a large computational burden. Two examples illustrate in computational terms of the accuracy and effectiveness of the proposed method.The authors are appreciative for anonymous referees for their hard work reading the paper and for their recommendations to improve the manuscript.info:eu-repo/semantics/publishedVersio