New exact solutions for seventh‐order Sawada–Kotera–Ito,Lax and Kaup–Kupershmidt equations using Exp‐function method

In this work, Exp‐function method is used to solve three different seventh‐order nonlinear partial differential KdV equations. Sawada–Kotera–Ito, Lax and Kaup–Kupershmidt equations are well known and considered for solve. Exp‐function method can be used as an alternative to obtain analytic and approximate solutions of different types of differential equations applied in engineering mathematics. Ultimately this method is implemented to solve these equations and convenient and effective solutions are obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Exact solutions for nonlinear partial differential equations via Exp‐function method

In this article, the Exp‐function method is applied to nonlinear Burgers equation and special fifth‐order partial differential equation. Using this method, we obtain exact solutions for these equations. The method is straightforward and concise, and its applications are promising. This method can be used as an alternative to obtain analytical and approximate solutions of different types of nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Application of Exp‐function method to EW‐Burgers equation

In this article, Exp‐function method is used to obtain an exact solution of the equal‐width wave‐Burgers equation (EW‐Burgers). The method is straightforward and concise, and its applications are promising. It is shown that Exp‐function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving EW‐Burgers equation. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A improved F‐expansion method and its application to Kudryashov–Sinelshchikov equation

On the basis of the F‐expansion method with a new sub‐equation and Exp‐function method, an improved F‐expansion method is introduced. As illustrative examples, the exact solutions expressed by exponential function, hyperbolic function of Kudryashov–Sinelshchikov equation for arbitrary α,β are derived. Some previous results are extended. The method is straightforward, concise and is a promising and powerful method for other nonlinear evolution equations in mathematical physics. Copyright © 2013 John Wiley & Sons, Ltd.     

Application of the exp‐function method to nonlinear lattice differential equations for multi‐wave and rational solutions

In this paper, we extend the basic Exp‐function method to nonlinear lattice differential equations for constructing multi‐wave and rational solutions for the first time. We consider a differential‐difference analogue of the Korteweg–de Vries equation to elucidate the solution procedure. Our approach is direct and unifying in the sense that the bilinear formalism of the equation studied becomes redundant. Copyright © 2011 John Wiley & Sons, Ltd.     

Homotopy perturbation method for numerical solutions of KdV‐Burgers' and Lax's seventh‐order KdV equations

In this article, we applied homotopy perturbation method to obtain the solution of the Korteweg‐de Vries Burgers (for short, KdVB) and Lax's seventh‐order KdV (for short, LsKdV) equations. The numerical results show that homotopy perturbation method can be readily implemented to this type of nonlinear equations and excellent accuracy. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A new approach by two‐dimensional wavelets operational matrix method for solving variable‐order fractional partial integro‐differential equations

In this paper, a new computational scheme based on operational matrices (OMs) of two‐dimensional wavelets is proposed for the solution of variable‐order (VO) fractional partial integro‐differential equations (PIDEs). To accomplish this method, first OMs of integration and VO fractional derivative (FD) have been derived using two‐dimensional Legendre wavelets. By implementing two‐dimensional wavelets approximations and the OMs of integration and variable‐order fractional derivative (VO‐FD) along with collocation points, the VO fractional partial PIDEs are reduced into the system of algebraic equations. In addition to this, some useful theorems are discussed to establish the convergence analysis and error estimate of the proposed numerical technique. Furthermore, computational efficiency and applicability are examined through some illustrative examples.     

A improved F‐expansion method and its application to the Zhiber–Shabat equation

Based on the F‐expansion method and Exp‐function method, an improved F‐expansion method is introduced. As illustrative examples, the exact solutions expressed by exponential function, hyperbolic functions, logarithmic function, and other type of functions for the Zhiber–Shabat equation are derived. Some previous results are extended. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

Solutions for the double Sine‐Gordon equation by Exp‐function,Tanh, and extended Tanh methods

In this work, we implement some analytical techniques such as the Exp‐function, Tanh, and extended Tanh methods for solving nonlinear partial differential equation, which contains sine terms, its name Double Sine‐Gordon equation. These methods obtain exact solutions of different types of differential equations in engineering mathematics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Second‐order Galerkin‐Lagrange method for the Navier‐Stokes equations (retracted article)

It has come to the attention of the editors and publisher that an article published in Numerical Methods and Partial Differential Equations, “Second‐order Galerkin‐Lagrange method for the Navier‐Stokes equations,” by Mohamed Bensaada, Driss Esselaoui, and Pierre Saramito, Numer Methods Partial Differential Eq 21(6) (2005), 1099–1121 included large portions that were copied from the following paper without proper citation: “Convergence and nonlinear stability of the Lagrange‐Galerkin method for the Navier‐Stokes equations,” Endre Suli, Numerische Mathematik, Vol. 53, No. 4, pp. 459–486 (July, 1988). We have retracted the paper and apologize to Dr. Suli Numer Methods Partial Differential Eq (2007)23(1)211 .     

Boundary‐value problems for the third‐order loaded equation with noncharacteristic type‐change boundaries

The purpose of this paper is to establish unique solvability for a certain generalized boundary‐value problem for a loaded third‐order integro‐differential equation with variable coefficients. Moreover, the method of integral equations is applied to obtain an equation related to the Riemann‐Liouville operators.     

Admissible meromorphic solutions of algebraic differential‐difference equations

By using Nevanlinna theory, we generalize a result given by Wittich to complex differential‐difference equations. The result obtained is that the differential‐difference equation in f which is of only one dominant term, has no admissible meromorphic solution f with hyper‐order less than 1 provided N(r,f) = S(r,f).     

Solvability of a nonlinear fifth‐order difference equation

Some formulas for well‐defined solutions to four very special cases of a nonlinear fifth‐order difference equation have been presented recently in this journal, where some of them were proved by the method of induction, some are only quoted, and no any theory behind the formulas was given. Here, we show in an elegant constructive way how the general solution to the difference equation can be obtained, from which the special cases very easily follow, which is also demonstrated here. We also give some comments on the local stability results on the special cases of the nonlinear fifth‐order difference equation previously publish in this journal.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite differences and collocation methods for the solution of the two‐dimensional heat equation

In this article, we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two‐dimensional heat equation. We employ, respectively, second‐order and fourth‐order schemes for the spatial derivatives, and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is nonsingular. Numerical experiments carried out on serial computers show the unconditional stability of the proposed method and the high accuracy achieved by the fourth‐order scheme. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 54–63, 2001     

Convergence analysis for second‐order accurate schemes for the periodic nonlocal Allen‐Cahn and Cahn‐Hilliard equations

In this paper, we provide a detailed convergence analysis for fully discrete second‐order (in both time and space) numerical schemes for nonlocal Allen‐Cahn and nonlocal Cahn‐Hilliard equations. The unconditional unique solvability and energy stability ensures ? ⁴ stability. The convergence analysis for the nonlocal Allen‐Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn‐Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an H ^?1 inner‐product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (eg, 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O (s ³+h ⁴) convergence in norm, in which s and h denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint s ≤C h . Here, we also prove convergence of the scheme in the maximum norm under the same constraint.     

Symmetry analysis of the nonlinear two‐dimensional Klein–Gordon equation with a time‐varying delay

The group analysis method is applied to the two‐dimensional nonlinear Klein–Gordon equation with time‐varying delay. Determining equations for equations with a time‐varying delay are derived. A complete group classification of the studied equation with respect to the function involved into the equation is obtained. All admitted Lie algebras are classified. By using the classifications, representations of all invariant solutions are found. Copyright © 2017 John Wiley & Sons, Ltd.