Integrated radial basis functions to simulate modified anomalous sub‐diffusion equation

In this article, a meshfree method for the numerical solution of conversation law equations is considered. Some problems which have shock such as advection problems are not properly solved by radial basis function collocation meshfree method. Therefore, we use the integrated radial basis function (IRBF) method for some of these problems. In the current study, the governing models have been discretized by IRBF technique in the spatial direction and by finite difference approximation for time variable. This converts the main problem to a system of nonlinear ordinary differential equations (ODEs). Furthermore, the obtained ODEs will be solved by Runge–Kutta technique. This is the meshless method of lines technique. Numerical examples indicate the acceptable accuracy, proficiency and easy implementation of the presented method.

Integrated multiquadric radial basis function approximation methods

Recently, the numerical modelling and simulation for anomalous subdiffusion equation (ASDE), which is a type of fractional partial differential equation( FPDE) and has been found with widely applications in modern engineering and sciences, are attracting more and more attentions. The current dominant numerical method for modelling ASDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of the non-linear ASDE. The discrete system of equations is obtained by using the meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formulations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the ASDE. Therefore, the meshless technique should have good potential in development of a robust simulation tool for problems in engineering and science which are governed by the various types of fractional differential equations.

<abstract><p>Fractional partial differential equations (PDEs) have key role in many physical, chemical, biological and economic problems. Different numerical techniques have been adopted to deal the multi-term FPDEs. In this article, the meshfree numerical scheme, Radial basis function (RBF) is discussed for some time-space fractional PDEs. The meshfree RBF method base on the Gaussian function and is used to test the numerical results of the time-space fractional PDE problems. Riesz fractional derivative and Grünwald-Letnikov fractional derivative techniques are used to deal the space fractional derivative terms while the time-fractional derivatives are iterated by Caputo derivative method. The accuracy of the suggested scheme is analyzed by using $ L_\infty $-norm. Stability and convergence analysis are also discussed.</p></abstract>

Direct meshless local Petrov–Galerkin (DMLPG) method for time-fractional fourth-order reaction–diffusion problem on complex domains

The trace-by-trace seismic deconvolution methods for reducing band-limited wavelet interference often cause lateral instability of the estimated reflectivity or impedance, and yield some trails like noodles, corresponding to relatively high wave-number components, when the signal-to-noise ratio of data is low and/or the data include high wave-number noise (or error). In this paper, a modified total variation (TV) regularization by linearly combining the sparseness regularization of model difference along the temporal direction and that along the spatial direction, where spatial regularization can be described as L1 norm of difference model along the interpreted horizontal direction, is introduced to develop a seismic deconvolution method to reduce even overcome these problems. The TV regularization connects the reflectivity at a point with its adjacent points and implicitly with any other points, thus introduces some additional useful information, especially for spatial continuity. In theory, minimizing TV has the ability to preserve edges of model parameters. A synthetic data example and a field data example are adopted to illustrate the performances of the method especially for the inherent role of sparseness regularization of model difference along the temporal and spatial directions.

A numerical scheme is developed to solve the time-fractional linear Kuramoto-Sivahinsky equation in this work. The time-fractional derivative (of order γ) is taken in the Caputo sense. The scheme comprises the backward Euler formula in the temporal direction and the quintic B-spline collocation approach in the spatial direction. Through rigorous analysis, the proposed method is shown to be unconditionally stable and convergent of order and two in the temporal and spatial directions, respectively. Two test problems are solved numerically to demonstrate the convergence and accuracy of the method.

Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modeling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is finite difference method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of FPDEs.

An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation

In this article, we consider a class of singularly perturbed mixed parabolic‐elliptic problems whose solutions possess both boundary and interior layers. To solve these problems, a hybrid numerical scheme is proposed and it is constituted on a special rectangular mesh which consists of a layer resolving piecewise‐uniform Shishkin mesh in the spatial direction and a uniform mesh in the temporal direction. The domain under consideration is partitioned into two subdomains. For the spatial discretization, the proposed scheme is comprised of the classical central difference scheme in the first subdomain and a hybrid finite difference scheme in the second subdomain, whereas the time derivative in the given problem is discretized by the backward‐Euler method. We prove that the method converges uniformly with respect to the perturbation parameter with almost second‐order spatial accuracy in the discrete supremum norm. Numerical results are finally presented to validate the theoretical results.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1931–1960, 2014

Generation of block-structured grids in complex computational domains using templates

Numerous fields, including the physical sciences, social sciences, and earth sciences, benefit greatly from the application of fractional calculus (FC). The fractional-order derivative is developed from the integer-order derivative, and in recent years, real-world modeling has performed better using the fractional-order derivative. Due to the flexibility of B-spline functions and their capability for very accurate estimation of fractional equations, they have been employed as a solution interpolating polynomials for the solution of fractional partial differential equations (FPDEs). In this study, cubic B-spline (CBS) basis functions with new approximations are utilized for numerical solution of third-order fractional differential equation. Initially, the CBS functions and finite difference scheme are applied to discretize the spatial and Caputo time fractional derivatives, respectively. The scheme is convergent numerically and theoretically as well as being unconditionally stable. On a variety of problems, the validity of the proposed technique is assessed, and the numerical results are contrasted with those reported in the literature.

A class of time‐dependent singularly perturbed convection‐diffusion problems with retarded terms arising in computational neuroscience is considered. In particular, a numerical scheme for the parabolic convection‐diffusion problem where the second‐order derivative with respect to the spatial direction is multiplied by a small perturbation parameter and the shifts are of is constructed. The Taylor series expansion is used to tackle the shift terms. The continuous problem is semidiscretized using the Crank‐Nicolson finite difference method in the temporal direction and the resulting set of ordinary differential equations is discretized using a midpoint upwind finite difference scheme on an appropriate piecewise uniform mesh, which is dense in the boundary layer region. It is shown that the proposed numerical scheme is second‐order accurate in time and almost first‐order accurate in space with respect to the perturbation parameter . To validate the computational results and efficiency of the method some numerical examples are encountered and the numerical results are compared with some existing results. It is observed that the numerical approximations are fairly good irrespective of the size of the delay and the advance till they are of . The effect of the shifts on the boundary layer has also been observed.

<sec>The time-resolved X-ray spectroscopy measurement system based on X-ray streak camera technology is indispensable diagnostic equipment in the study of laser inertial fusion research and high-energy-density physics. However, limited by the effective photocathode length of the X-ray streak tube, the time-resolved spectral measurement system usually used has the shortcomings of narrow spectrum range and poor spectral resolution.</sec><sec>In order to overcome the shortcomings, a novel dual-channel streak tube is developed, which consists of a photocathode, a prefocusing electrode group in temporal direction, an electric quadrupole lens electrode group, a main focusing electrode group in temporal direction, a deflector plate, and a phosphor screen. The photocathode has two slits. When X-rays are incident, two electron beams can be emitted simultaneously. The electric quadrupole lens electrode group is composed of 8 arc electrodes. Two electric quadrupole lenses are formed by the 8 arc electrodes in the spatial direction. Two electron beams emitted from the cathode of the streak tube are first accelerated and prefocused by the prefocusing electrode group in the time direction, and then compressed by the main focusing electrode group in the time direction. In the spatial direction, two electron beams are focused by the two electric quadrupole lenses independently. This novel streak tube structure can focus two electron beams at the same time, thereby increasing the effective photocathode length and maintaining the compact structure of streak tube without increasing the aberration.</sec><sec>The cathode voltage of the designed streak tube is –12 kV, the distance from cathode to grid is 5 mm, and the cathode-grid field strength is 2.4 kV/mm. The cathode is divided into two sections, the spacing between sections is about 13 mm, the length of each section is more than 20 mm, the magnification of the image converter tube is about 1.56 times, the distance between the cathode and the phosphor screen is 300 mm, and the longest size along the cathode direction is 90 mm. The test results of the performance of the streak tube show that the actual effective cathode length of the developed tube reaches 44 mm, the spatial resolution is better than 15 lp/mm, and the deflection sensitivity is better than 40 mm/kV. The effective cathode and spatial resolution of the tube can be increased to 50 mm and 25 lp/mm by further optimizing the structure of the tube and removing the image intensifier with a high sensitivity image recording system, respectively.</sec>

ABSTRACTIn this paper a finite difference method is presented to solve time–space linear and nonlinear fractional diffusion equations. Specifically, the centred difference scheme is used to approximate the Riesz fractional derivative in space. A trapezoidal formula is used to solve a system of Volterra integral equations transformed from spatial discretization. Stability and convergence of the proposed scheme is discussed which shows second-order accuracy both in temporal and spatial directions. Finally, examples are presented to show the accuracy and effectiveness of the schemes.