A fast algorithm for solving the space–time fractional diffusion equation

In this paper, we propose a fast algorithm for efficient and accurate solution of the space–time fractional diffusion equations defined in a rectangular domain. The spatial discretization is done by using the central finite difference scheme and matrix transfer technique. Due to its nonlocality, numerical discretization of the spectral fractional Laplacian ${(?\Delta )}_s^{\alpha /2}$ results in a large dense matrix. This causes considerable challenges not only for storing the matrix but also for computing matrix–vector products in practice. By utilizing the compact structure of the discrete system and the discrete sine transform, our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs. We then use the Laplace transform method for time integration of the semi-discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme. Various experiments are presented to demonstrate the efficiency and accuracy of our method.     

An efficient parallel solution for Caputo fractional reaction–diffusion equation

The computational complexity of Caputo fractional reaction–diffusion equation is \(O(MN^2)\) compared with \(O(MN)\) of traditional reaction–diffusion equation, where \(M\) , \(N\) are the number of time steps and grid points. A efficient parallel solution for Caputo fractional reaction–diffusion equation with explicit difference method is proposed. The parallel solution, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector vector addition and constant vector multiplication. The sum of constant vector multiplication is optimized. As to the authors’ knowledge, this is the first parallel solution for Caputo fractional reaction–diffusion equation. The experimental results show that the parallel solution compares well with the analytic solution. The parallel solution on single Intel Xeon X5540 CPU runs more than three times faster than the serial solution on single X5540 CPU core, and scales quite well on a distributed memory cluster system.     

A wavelet Petrov–Galerkin method for solving integro-differential equations

In this paper, we solve integro-differential equation by using the Alpert multiwavelets as basis functions. We also use the orthogonality of the basis of the trial and test spaces in the Petrov–Galerkin method. The computations are reduced because of orthogonality. Thus the final system that we get from discretizing the integro-differential equation has a very small dimension and enough accuracy. We compare the results with [M. Lakestani, M. Razzaghi, and M. Dehghan, Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations, Math. Probl. Eng. 2006 (2006), pp. 1–12, Article ID 96184] and [A. Ayad, Spline approximation for first-order Fredholm integro-differential equation, Stud. Univ. Babes-Bolyai. Math., 41(3), (1996), pp. 1–8] which used a much larger dimension system and got less accurate results. In [Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind integral equations, SIAM J. Numer. Anal. 35(1) (1998), pp. 406–434], convergence of Petrov–Galerkin method has been discussed with some restrictions on degrees of chosen polynomial basis, but in this paper convergence is obtained for every degree.     

Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

Engineering with Computers - This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational...     

A meshless method for solving the time fractional advection–diffusion equation with variable coefficients

In this paper, an efficient and accurate meshless method is proposed for solving the time fractional advection–diffusion equation with variable coefficients which is based on the moving least square (MLS) approximation. In the proposed method, firstly the time fractional derivative is approximated by a finite difference scheme of order $O\left({\left(\delta t\right)}^{2?\alpha }\right),0<\alpha \le 1$ and then the MLS approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the Caputo sense. Also, the validity of the proposed method is investigated in error analysis discussion. The main aim is to show that the meshless method based on the MLS shape functions is highly appropriate for solving fractional partial differential equations (FPDEs) with variable coefficients. The efficiency and accuracy of the proposed method are verified by solving several examples.     

A bounded numerical solver for a fractional FitzHugh–Nagumo equation and its high-performance implementation

In this paper, we propose a high-performance implementation of a space-fractional FitzHugh–Nagumo model. Our implementation is based on a positivity- and boundedness-preserving finite-difference model to approximate the solutions of a Riesz space-fractional reaction-diffusion equation. The model generalizes the FitzHugh–Nagumo model. The stability and convergence of the difference scheme are thoroughly discussed. Moreover, we prove the existence and uniqueness of numerical solutions, positivity, boundedness and consistency of the model. The scheme is based on weighted and shifted Grünwald differences. The conjugate gradient method is used then to solve the sparse matrix system. The MPI and PETSc libraries are used for the computational implementation. We investigate the influence of some computer factors on the performance of our implementation and scalability. More precisely, we consider the number of cores, the size of the computation mesh and the orders of the fractional derivatives. Tests are evaluated on a ccNUMA architecture with two CPUs.

     

Fourth-order accurate difference schemes for solving Benjamin–Bona–Mahony–Burgers (BBMB) equation

Engineering with Computers - In this paper, two high-order difference schemes for the Benjamin–Bona–Mahony–Burgers (BBMB) equation are proposed. The first scheme is two level and...     

A novel time-marching scheme using numerical Green’s functions: A comparative study for the scalar wave equation

The present paper presents the formulation of a novel time-marching method based on the Explicit Green’s Approach (ExGA) to solve scalar wave propagation problems. By means of the weighted residual method in both time and space, the time integral expression concerning the ExGA is readily established. The arising ExGA time integral expression is spatially discretized in a finite element sense and a recursive scheme that employs time-domain numerical Green’s function matrices is adopted to evaluate the displacement and the velocity vectors. These Green’s matrices are computed by the time discontinuous Galerkin finite element method only at the first time step. The system of coupled equations originated from the time discontinuous Galerkin method is then solved by an iterative predictor–multicorrector algorithm. Once the Green’s matrices are computed, no iterative process is required to obtain the displacement and the velocity vectors at any time level. At the end of the paper, numerical examples are presented in order to compare the proposed approach with other approaches.     

A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations

In this paper, to find an approximate solution of general linear Fredholm integro-differential–difference equations (FIDDEs) under the initial-boundary conditions in terms of the Bessel polynomials, a practical matrix method is presented. The idea behind the method is that it converts FIDDEs to a matrix equation which corresponds to a system of linear algebraic equations and is based on the matrix forms of the Bessel polynomials and their derivatives by means of collocation points. The solutions are obtained as the truncated Bessel series in terms of the Bessel polynomials J _n (x) of the first kind defined in the interval [0, ∞). The error analysis and the numerical examples are included to demonstrate the validity and applicability of the technique.     

Linearized Crank–Nicolson method for solving the nonlinear fractional diffusion equation with multi-delay

This paper is concerned with numerical solution of the nonlinear fractional diffusion equation with multi-delay. The studied model plays a significant role in population ecology. A linearized Crank–Nicolson method for such problem is proposed by combing the Crank–Nicolson approximation in time with the fractional centred difference formula in space. Using the discrete energy method, the suggested scheme is proved to be uniquely solvable, stable and convergent with second-order accuracy in both space and time for sufficiently small space and time increments. Several numerical experiments for solving the delay fractional Hutchinson equation and two real problems in population dynamics are provided to verify our theoretical results.     

The numerical integration scheme for a fast Petrov–Galerkin method for solving the generalized airfoil equation

The main purpose of this paper is to give the numerical integration scheme for a fast Petrov–Galerkin method for solving the generalized airfoil equation, considered in a recent paper (Cai, J. Complex. 25:420–436, 2009). This scheme leads to a fully discrete sparse linear system. We show that it requires a nearly linear computational cost to get this system, and the approximate solution of the resulting linear system preserves the optimal convergent order. Numerical experiments are presented to confirm the theoretical estimates.     

A family of fourth-order difference schemes on rotated grid for two-dimensional convection–diffusion equation

We derive a family of fourth-order finite difference schemes on the rotated grid for the two-dimensional convection–diffusion equation with variable coefficients. In the case of constant convection coefficients, we present an analytic bound on the spectral radius of the line Jacobi’s iteration matrix in terms of the cell Reynolds numbers. Our analysis and numerical experiments show that the proposed schemes are stable and produce highly accurate solutions. Classical iterative methods with these schemes are convergent with large values of the convection coefficients. We also compare the fourth-order schemes with the nine point scheme obtained from the second-order central difference scheme after one step of cyclic reduction.     

A numerical study of stationary solution of viscous Burgers’ equation using wavelet

In this paper we have studied the numerical stationary solution of viscous Burgers’ equation with Neumann boundary conditions by applying wavelet Galerkin method. Burns et al. [J. Burns, A. Balogh, D.S. Gilliam, and V. I. Shubov, Numerical stationary solutions for a viscous Burgers’ equation, J. Maths. Sys. Est. Contl. 8 (1998), pp. 1–16] have reported that for moderately small viscosity and for certain initial conditions, numerical solution approaches non-constant shock-type stationary solution though only possible actual stationary solution is a constant. We found that the wavelet Galerkin method precisely captures the correct steady-state solution. The solutions obtained were impressive and verify theoretical results.     

Compact difference method for solving the fractional reaction–subdiffusion equation with Neumann boundary value condition

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with order O(τ^2?α+h⁴) in L₂ norm, where τ, α, and h are the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme.     

A Galerkin finite element scheme for time–space fractional diffusion equation

In this paper, a Galerkin finite element scheme to approximate the time–space fractional diffusion equation is studied. Firstly, the fractional diffusion equation is transformed into a fractional Volterra integro-differential equation. And a second-order fractional trapezoidal formula is used to approach the time fractional integral. Then a Galerkin finite element method is introduced in space direction, where the semi-discretization scheme and fully discrete scheme are given separately. The stability analysis of semi-discretization scheme is discussed in detail. Furthermore, convergence analysis of semi-discretization scheme and fully discrete scheme are given in details. Finally, two numerical examples are displayed to demonstrate the effectiveness of the proposed method.     

A haar wavelet approximation for two-dimensional time fractional reaction–subdiffusion equation

Engineering with Computers - In this study, we established a wavelet method, based on Haar wavelets and finite difference scheme for two-dimensional time fractional reaction–subdiffusion...     

A numerical study of the Navier–Stokes-αβ model

We present a numerical study of the NS-αβ model, which is a recently proposed multiscale variation of the NS-α model that attempts to recapture scales lost through over-regularization by separately modeling dissipation-range scales. We develop a similarity theory for the new model which shows that it is better equipped than the NS-α model to capture smaller-scale behavior. Next, we propose and study an unconditionally stable, optimally accurate, and efficient finite-element implementation for the NS-αβ model; rigorous proofs for stability and convergence are provided. Finally, we present results from two numerical experiments that demonstrate the advantages of the NS-αβ model over the NS-α model.     

A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation

In this paper, an efficient numerical scheme based on uniform Haar wavelets and the quasilinearization process is proposed for the numerical simulation of time dependent nonlinear Burgers’ equation. The equation has great importance in many physical problems such as fluid dynamics, turbulence, sound waves in a viscous medium etc. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. More accurate solutions are obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents a solution of boundary value problems. The accuracy of the proposed method is demonstrated by three test problems. The numerical results are compared with existing numerical solutions found in the literature. The use of the uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and has small computation costs.     

Ket–Bra entangled state method for solving master equation of finite-level system

In this paper, we first introduce Ket–Bra entangled state method to solve master equation of finite-level system, which can convert master equation into Schrödinger-like equation and solve it with the mature methodology of Schrödinger equation. Then, several physical models include a radioactivity damped 2-level atom driven by classical field, a J–C model with cavity damping, a V-type qutrit under amplitude damping and N-qubits open Heisenberg chain have been solved with KBES method. Furthermore, the dynamic evolution and decoherence process of these models are investigated.     

Stability of standing waves for the fractional Schrödinger–Choquard equation

In this paper, we consider the stability of standing waves for the fractional Schrödinger–Choquard equation with an $L^2$-critical nonlinearity. By using the profile decomposition of bounded sequences in $H^s$ and variational methods, we prove that the standing waves are orbitally stable. We extend the study of Bhattarai for a single equation (Bhattarai, 2017) to the $L^2$-critical case.