A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients

The aim of this paper is to present a new numerical method for solving a wide class of fractional partial differential equations (FPDEs) such as wave-diffusion equations, modified anomalous fractional sub-diffusion equations, time-fractional telegraph equations. The proposed method is based on the Fourier series expansion along the spatial coordinate which transforms the original equation into a sequence of multi-term fractional ordinary differential equations (ODEs). These fractional equations are solved by the use of a new efficient numerical technique – the backward substitution method. The numerical examples confirm the high accuracy and efficiency of the proposed numerical scheme in solving FPDEs with variable in time coefficients.     

Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on each subinterval. The method is shown to be unconditionally stable, and for general nonlinear equations, the uniform sharp numerical order 3 − $ν$ can be rigorously proven for sufficiently smooth solutions at all time steps. The proof provides a general guide for proving the sharp order for higher-order schemes in the nonlinear case. Some numerical examples are given to validate our theoretical results.     

A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS

In this paper, an efficient numerical method for solving the general fractional diffusion equations with Riesz fractional derivative is proposed by combining the fractional compact difference operator and the boundary value methods. In order to efficiently solve the generated linear large-scale system, the generalized minimal residual (GMRES) algorithm is applied. For accelerating the convergence rate of the iterative, the Strang-type, Chan-type and P-type preconditioners are introduced. The suggested method can reach higher order accuracy both in space and in time than the existing methods. When the used boundary value method is $A_{k1,k2}$-stable, it is proven that Strang-type preconditioner is invertible and the spectra of preconditioned matrix is clustered around 1. It implies that the iterative solution is convergent rapidly. Numerical experiments with the absorbing boundary condition and the generalized Dirichlet type further verify the efficiency.     

Existence of Solutions to a Class of Fractional Differential Equations

In this paper, the existence of solutions to a class of fractional differential equations $D_{0+}^{\alpha}u(t)=h(t)f(t, u(t), D_{0+}^{\theta}u(t))$ is obtained by an efficient and simple monotone iteration method. At first, the existence of a solution to the problem above is guaranteed by finding a bounded domain $D_M$ on functions $f$ and $g$. Then, sufficient conditions for the existence of monotone solution to the problem are established by applying monotone iteration method. Moreover, two efficient iterative schemes are proposed, and the convergence of the iterative process is proved by using the monotonicity assumption on $f$ and $g$. In particular, a new algorithm which combines Gauss-Kronrod quadrature method with cubic spline interpolation method is adopted to achieve the monotone iteration method in Matlab environment, and the high-precision approximate solution is obtained. Finally, the main results of the paper are illustrated by some numerical simulations, and the approximate solutions graphs are provided by using the iterative method.     

Third order accurate variable mesh discretization and application of TAGE iterative method for the non-linear two-point boundary value problems with homogeneous functions in integral form

In this paper, we discuss the application of two-parameter alternating group explicit (TAGE) iterative method to an efficient third order variable mesh method for the solution of non-linear differential equations with integral homogeneous functions subject to natural boundary conditions. The proposed method is applicable only when the internal grid points of the solution space are odd in number. The proposed iterative method is also applicable to the integro-differential equations with singular coefficients. Comparative numerical results are given to demonstrate the usefulness of the proposed method.     

A simple finite volume method for the shallow water equations

We present a new finite volume method for the numerical solution of shallow water equations for either flat or non-flat topography. The method is simple, accurate and avoids the solution of Riemann problems during the time integration process. The proposed approach consists of a predictor stage and a corrector stage. The predictor stage uses the method of characteristics to reconstruct the numerical fluxes, whereas the corrector stage recovers the conservation equations. The proposed finite volume method is well balanced, conservative, non-oscillatory and suitable for shallow water equations for which Riemann problems are difficult to solve. The proposed finite volume method is verified against several benchmark tests and shows good agreement with analytical solutions.     

Unconditionally Stable Splitting Methods for the Shallow Water Equations

The front-tracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The front-tracking method is unconditionally stable, but for practical computations feasible CFL numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is highly efficient. The numerical technique is applied to four test cases, the first being an expanding bore with rotational symmetry. The second problem addresses the question of describing the time development of two constant water levels separated by a dam that breaks instantaneously. The third problem compares the front-tracking method with an explicit analytic solution of water waves rotating over a parabolic bottom profile. Finally, we study flow over an obstacle in one dimension.     

Operator splitting methods for pricing American options under stochastic volatility

We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.     

The fractional steps domain decomposition method for numerical solution of a class of viscous wave equations

In this article, an efficient fractional steps domain decomposition method (FSDDM) is derived for parallel numerical solution of a class of viscous wave equations. In this procedure, the large domain is divided into multiple block sub-domains. The values on the interfaces of sub-domains are found by an efficient local multilevel scheme, implicit scheme is used for computing the interior values in sub-domains. Some techniques, such as non-overlapping domain decomposition, fractional steps and extrapolation algorithm are adopted. Numerical experiments are performed to demonstrate the efficiency and accuracy of the method.     

Numerical model for the shallow water equations on a curvilinear grid with the preservation of the Bernoulli integral

Methodological aspects concerning the construction of a two-dimensional numerical model for reservoir flows based on the shallow water equations are considered. A numerical scheme is constructed by applying the control volume method on staggered grids in combination with the Bernoulli integral, which is used to interpolate the desired fields inside a grid cell. The implementation of the method yields a monotone numerical scheme. The results of numerical integration are compared with the exact solution.     

An iterative algorithm for solving a class of generalized coupled Sylvester-transpose matrix equations over bisymmetric or skew-anti-symmetric matrices

This paper presents an iterative algorithm to solve a class of generalized coupled Sylvester-transpose matrix equations over bisymmetric or skew-anti-symmetric matrices. When the matrix equations are consistent, the bisymmetric or skew-anti-symmetric solutions can be obtained within finite iteration steps in the absence of round-off errors for any initial bisymmetric or skew-anti-symmetric matrices by the proposed iterative algorithm. In addition, we can obtain the least norm solution by choosing the special initial matrices. Finally, numerical examples are given to demonstrate the iterative algorithm is quite efficient. The merit of our method is that it is easy to implement.     

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

In this paper, an efficient and accurate numerical method is presented for solving two types of fractional partial differential equations. The fractional derivative is described in the Caputo sense. Our approach is based on Bernoulli wavelets collocation techniques together with the fractional integral operator, described in the Riemann‐Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved and the solution of fractional partial differential equations is achieved. Some results concerning the error analysis are obtained. The validity and applicability of the method are demonstrated by solving four numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions much easier.     

求解二维浅水波方程的移动网格旋转通量法

为提高求解二维浅水波方程数值算法的分辨率,拟构造求解该方程的新算法:基于移动网格法,选用熵稳定数值通量函数,利用旋转不变性得到混合数值通量.该算法中,浅水波方程的数值求解和依据解的特性进行自适应疏密分布的网格计算过程交错进行.利用变分原理进行网格重构,新网格上的物理量采用二阶精度的守恒型插值公式计算,最终采用三阶强稳定Runge-Kutta法与满足热力学第二定律的熵稳定格式实现浅水波方程的数值求解.数值结果表明,新算法具有良好的间断捕捉能力,分辨率高.     

A new method for exact solutions of variant types of time‐fractional Korteweg‐de Vries equations in shallow water waves

The current article devoted on the new method for finding the exact solutions of some time‐fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time‐fractional equations viz. time‐fractional KdV‐Burgers and KdV‐mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

一类分数阶Langevin方程block-by-block算法的数值分析

分数阶Langevin方程有重要的科学意义和工程应用价值,基于经典block-by-block算法,求解了一类含有Caputo导数的分数阶Langevin方程的数值解.Block-by-block算法通过引入二次Lagrange基函数插值,构造出逐块收敛的非线性方程组,通过在每一块耦合求得分数阶Langevin方程的数值解.在0<α<1条件下,应用随机Taylor展开证明block-by-block算法是3+α阶收敛的,数值试验表明在不同α和时间步长h取值下,block-by-block算法具有稳定性和收敛性,克服了现有方法求解分数阶Langevin方程速度慢精度低的缺点,表明block-by-block算法求解分数阶Langevin方程是高效的.     

Application of Splitting Algorithms in the Method of Finite Volumes for Numerical Solution of the Navier–Stokes Equations

We generalize the splitting algorithms proposed earlier for the construction of efficient difference schemes to the finite volume method. For numerical solution of the Euler and Navier–Stokes equations written in integral form, some implicit finite-volume predictor-corrector scheme of the second order of approximation is proposed. At the predictor stage, the introduction of various forms of splitting is considered, which makes it possible to reduce the solution of the original system to separate solution of individual equations at fractional steps and to ensure some stability margin of the algorithm as a whole. The algorithm of splitting with respect to physical processes and spatial directions is numerically tested. The properties of the algorithm are under study and we proved its effectiveness for solving two-dimensional and three-dimensional flow-around problems.     

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for . The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation.     

基于拟Shannon小波浅水长波近似方程组的数值解

本文研究了浅水长波近似方程组初边值问题的数值解.利用小波多尺度分析和区间拟Shannon小波,对浅水长波近似方程组空间导数实施空间离散,用时间步长自适应精细积分法对其变换所的非线性常微分方程组进行求解,得到了浅水长波近似方程组的数值解,并将此方法计算的结果与其解析解进行比较和验证.     

Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order

In order to control the movement of waves on the area of shallow water, the newly derivative with fractional order proposed by Caputo and Fabrizio was used. To achieve this, we first proposed a transition from ordinary to fractional differential equation. We proved the existence and uniqueness of the coupled solutions of the modified system using the fixed-point theorem. We derive the special solution of the modified system using an iterative method. We proved the stability of the used method and also the uniqueness of the special solution. We presented the numerical simulations for different values of alpha.     

Numerical simulation of the fractional-order control system

Multi-term fractional differential equations have been used to simulate fractional-order control system. It has been demonstrated the necessity of the such controllers for the more efficient control of fractionalorder dynamical system. In this paper, the multi-term fractional ordinary differential equations are transferred into equivalent a system of equations. The existence and uniqueness of the new system are proved. A fractional order difference approximation is constructed by a decoupled technique and fractional-order numerical techniques. The consistence, convergence and stability of the numerical approximation are proved. Finally, some numerical results are presented to demonstrate that the numerical approximation is a computationally efficient method. The new method can be applied to solve the fractional-order control system.