Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

In this article, the generalized Rosenau–KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie–Trotter and the second‐order Strang time‐splitting techniques combined with the quintic B‐spline collocation by the help of the fourth order Runge–Kutta (RK‐4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L₂ and L_∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.     

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015     

Spectral collocation method and Darvishi’s preconditionings to solve the generalized Burgers–Huxley equation

In this paper, a numerical solution of the generalized Burgers–Huxley equation is presented. This is the application of spectral collocation method. To reduce roundoff error in this method we use Darvishi’s preconditionings. The numerical results obtained by this method have been compared with the exact solution. It can be seen that they are in a good agreement with each other, because errors are very small and figures of exact and numerical solutions are very similar.     

Discrete Jacobi sub‐equation method for nonlinear differential–difference equations

We will propose a unified algebraic method to construct Jacobi elliptic function solutions to differential–difference equations (DDEs). The solutions to DDEs in terms of Jacobi elliptic functions sn, cn and dn have a unified form and can be presented through solving the associated algebraic equations. To illustrate the effectiveness of this method, we apply the algorithm to some physically significant DDEs, including the discrete hybrid equation, semi‐discrete coupled modified Korteweg–de Vries and the discrete Klein–Gordon equation, thereby generating some new exact travelling periodic solutions to the discrete Klein–Gordon equation. A procedure is also given to determine the polynomial expansion order of Jacobi elliptic function solutions to DDEs. Copyright © 2010 John Wiley & Sons, Ltd.     

A robust semi-explicit difference scheme for the Kuramoto–Tsuzuki equation

In this paper, we propose a robust semi-explicit difference scheme for solving the Kuramoto–Tsuzuki equation with homogeneous boundary conditions. Because the prior estimate in L_∞-norm of the numerical solutions is very hard to obtain directly, the proofs of convergence and stability are difficult for the difference scheme. In this paper, we first prove the second-order convergence in L₂-norm of the difference scheme by an induction argument, then obtain the estimate in L_∞-norm of the numerical solutions. Furthermore, based on the estimate in L_∞-norm, we prove that the scheme is also convergent with second order in L_∞-norm. Numerical examples verify the correction of the theoretical analysis.     

A three‐level linearized finite difference scheme for the camassa–holm equation

The Camassa–Holm (CH) system is a strong nonlinear third‐order evolution equation. So far, the numerical methods for solving this problem are only a few. This article deals with the finite difference solution to the CH equation. A three‐level linearized finite difference scheme is derived. The scheme is proved to be conservative, uniquely solvable, and conditionally second‐order convergent in both time and space in the discrete L_∞ norm. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 451–471, 2014     

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

This article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction is O(dt) . Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss–Legendre–Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the one‐ and two‐dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples.     

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L₂, L_∞ and three invariants I₁, I₂, and I₃ are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.     

A Petrov–Galerkin spectral method for fourth‐order problems

In this paper, we consider the Petrov–Galerkin spectral method for fourth‐order elliptic problems on rectangular domains subject to non‐homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher‐order problems. By applying these results to a fourth‐order problem, we establish the H²‐error and L²‐error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Lp – Lq Decay Estimates fo the Solutions of Strictly Hyperbolic Equations of Second Order with Increasing in Time Coefficients

One of the most important questions in the theory of nonlinear wave equations is that for global existence of solutions. An essential tool is the Strichartz inequality for special solutions of the wave equation.In the last time different results were proved generalizing the classical one of Strichartz. In the present paper L_p – L_q estimates are proved for the solutions of strictly hyperbolic equations of second order with time dependent coefficients where these are unbounded at infinity. In the first step the WKB method is applied to the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. In a second step the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter.     

Stability of the Crank–Nicolson–Adams–Bashforth scheme for the 2D Leray‐alpha model

We consider the stability of an efficient Crank–Nicolson–Adams–Bashforth method in time, finite element in space, discretization of the Leray‐α model. We prove finite‐time stability of the scheme in L², H¹, and H², as well as the long‐time L‐stability of the scheme under a Courant‐Freidrichs‐Lewy (CFL)‐type condition. Numerical experiments are given that are in agreement with the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1155–1183, 2016     

A compact finite difference method for the solution of the generalized Burgers–Fisher equation

In this article, numerical solutions of the generalized Burgers–Fisher equation are obtained using a compact finite difference method with minimal computational effort. To verify this, a combination of a sixth‐order compact finite difference scheme in space and a low‐storage third‐order total variation diminishing Runge–Kutta scheme in time have been used. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. The approximate solutions to the equation have been computed without transforming the equation and without using linearization. Comparisons indicate that there is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy. The present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Navier–Stokes equations with slip boundary conditions

In this paper, we consider incompressible viscous fluid flows with slip boundary conditions. We first prove the existence of solutions of the unsteady Navier–Stokes equations in n‐spacial dimensions. Then, we investigate the stability, uniqueness and regularity of solutions in two and three spacial dimensions. In the compactness argument, we construct a special basis fulfilling the incompressibility exactly, which leads to an efficient and convergent spectral method. In particular, we avoid the main difficulty for ensuring the incompressibility of numerical solutions, which occurs in other numerical algorithms. We also derive the vorticity‐stream function form with exact boundary conditions, and establish some results on the existence, stability and uniqueness of its solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

A superconvergent nonconforming mixed finite element method for the Navier–Stokes equations

The superconvergence for a nonconforming mixed finite element approximation of the Navier–Stokes equations is analyzed in this article. The velocity field is approximated by the constrained nonconforming rotated Q₁ (CNRQ₁) element, and the pressure is approximated by the piecewise constant functions. Under some regularity assumptions, the superconvergence estimates for both the velocity in broken H¹‐norm and the pressure in L²‐norm are obtained. Some numerical examples are presented to demonstrate our theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 646–660, 2016     

Convergence analysis of a linearized Crank–Nicolson scheme for the two‐dimensional complex Ginzburg–Landau equation

A linearized Crank–Nicolson‐type scheme is proposed for the two‐dimensional complex Ginzburg–Landau equation. The scheme is proved to be unconditionally convergent in the L² ‐norm by the discrete energy method. The convergence order is \begin{align*}\mathcal{O}(\tau^2+h_1^2+h^2_2)\end{align*}, where τ is the temporal grid size and h₁,h₂ are spatial grid sizes in the x ‐ and y ‐directions, respectively. A numerical example is presented to support the theoretical result. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Global C∞‐solutions to 1D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we consider the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries. The initial density ρ₀∈W^1,2n is bounded below away from zero and the initial velocity u₀∈L²ⁿ. The viscosity coefficient µ is proportional to ρ^θ with 0<θ?1, where ρis the density. The existence and uniqueness of global solutions in Hⁱ([0,1])(i = 1,2,4) have been established in (J. Math. Phys. 2009; 50 :023101; Meth. Appl. Anal. 2005; 12 :239–252; J. Differ. Equations 2008; 245:3956–3973; Commun. Pure Appl. Anal. 2008; 7 :373–381). By mathematical induction method, we will establish the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries when the initial data ρ₀ and u₀ are smooth. Copyright © 2011 John Wiley & Sons, Ltd.     

Energy decay estimates for the dissipative wave equation with space–time dependent potential

We study the asymptotic behavior of solutions of dissipative wave equations with space–time‐dependent potential. When the potential is only time‐dependent, Fourier analysis is a useful tool to derive sharp decay estimates for solutions. When the potential is only space‐dependent, a powerful technique has been developed by Todorova and Yordanov to capture the exact decay of solutions. The presence of a space–time‐dependent potential, as in our case, requires modifications of this technique. We find the energy decay and decay of the L² norm of solutions in the case of space–time‐dependent potential. Copyright © 2010 John Wiley & Sons, Ltd.     

Littlewood–Paley Theory and Function Spaces with Alocp Weights

Characterizations via convolutions with smooth compactly supported kernels and other distinguished properties of the weighted Besov–Lipschitz and Triebel–Lizorkin spaces on ℝⁿ with weights that are locally in A_p but may grow or decrease exponentially at infinity are investigated. Square–function characterizations of the weighted L^p and Hardy spaces with the above class of weights are also obtained. A certain local variant of the Calderón reproducing formula is constructed and widely used in the proofs.     

A fast second‐order difference scheme for the space–time fractional equation

In this paper, a fast second‐order accurate difference scheme is proposed for solving the space–time fractional equation. The temporal Caputo derivative is approximated by ?L2 ‐1_σ formula which employs the sum‐of‐exponential approximation to the kernel function appeared in Caputo derivative. The second‐order linear spline approximation is applied to the spatial Riemann–Liouville derivative. At each time step, a fast algorithm, the preconditioned conjugate gradient normal residual method with a circulant preconditioner (PCGNR), is used to solve the resulting system that reduces the storage and computational cost significantly. The unique solvability and unconditional convergence of the difference scheme are shown by the discrete energy method. Numerical examples are given to verify numerical accuracy and efficiency of the difference schemes.     

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L²(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L²(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H¹(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.