Nonautonomous first integrals and general solutions of the KdV‐Burgers and mKdV‐Burgers equations with the source

The method for constructing first integrals and general solutions of nonlinear ordinary differential equations is presented. The method is based on index accounting of the Fuchs indices, which appeared during the Painlevé test of a nonlinear differential equation. The Fuchs indices indicate us the leading members of the first integrals for the origin differential equation. Taking into account the values of the Fuchs indices, we can construct the auxiliary equation, which allows to look for the first integrals of nonlinear differential equations. The method is used to obtain the first integrals and general solutions of the KdV‐Burgers and the mKdV‐Burgers equations with a source. The nonautonomous first integrals in the polynomials form are found. The general solutions of these nonlinear differential equations under at some additional conditions on the parameters of differential equations are also obtained. Illustrations of some solutions of the KdV‐Burgers and the mKdV‐Burgers are given.     

Conditional Lie‐Bäcklund Symmetry of Evolution System and Application for Reaction‐Diffusion System

Conditional Lie‐Bäcklund symmetry (CLBS) method is developed to study system of evolution equations. It is shown that reducibility of a system of evolution equations to a system of ordinary differential equations can be fully characterized by the CLBS of the considered system. As an application of the approach, a class of two‐component nonlinear diffusion equations is studied. The governing system and the admitted CLBS can be identified. As a consequence, exact solutions defined on the polynomial, exponential, trigonometric, and mixed invariant subspaces are constructed due to the corresponding symmetry reductions.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

Exact solutions of nonlinear time fractional partial differential equations by sub‐equation method

In this article, the sub‐equation method is presented for finding the exact solutions of a nonlinear fractional partial differential equations. For this, the fractional complex transformation method has been used to convert fractional‐order partial differential equation to ordinary differential equation. The fractional derivatives are described in Jumarie's the modified Riemann–Liouville sense. We apply to this method for the nonlinear time fractional differential equations. With the aid of symbolic computation, a variety of exact solutions for them are obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

On the concept of exact solution for nonlinear differential equations of fractional‐order

In this article, the new exact travelling wave solutions of the nonlinear space‐time fractional Burger's, the nonlinear space‐time fractional Telegraph and the nonlinear space‐time fractional Fisher equations have been found. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into ordinary differential equations of integer order in the sense of the Jumarie's modified Riemann–Liouville derivative. The ‐expansion method is effective for constructing solutions to the nonlinear fractional equations, and it appears to be easier and more convenient by means of a symbolic computation system. Copyright © 2016 John Wiley & Sons, Ltd.     

Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by homotopy analysis method

The similarity transform for the steady three-dimensional problem of a condensation film on an inclined rotating disk gives a system of nonlinear ordinary differential equations which are analytically solved by applying a newly developed method namely the homotopy analysis method (HAM). The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. The velocity and temperature profiles are shown and the influence of the Prandtl number on the heat transfer and the Nusselt number is discussed in detail. The validity of our results is verified by numerical results.     

Flow of a Reiner‐Rivlin fluid between two infinite coaxial rotating disks

Present study deals with the steady flow and heat transfer of a non‐Newtonian Reiner‐Rivlin fluid between two coaxially rotating infinite disks. Using similarity transformations, the governing equations are reduced to a set of nonlinear, highly coupled ordinary differential equations and by means of an effective analytical method called homotopy analysis method; analytical solutions are constructed in series form. Different cases, such as, when one disk is at rest and the other is rotating with constant angular velocity, two disks rotating with different angular velocities in same as well as opposite sense, two disks rotating with same angular velocities in opposite sense, are discussed. The effects of non‐Newtonian parameter, Reynolds number, are also discussed, and results are presented graphically.     

Using empirical Eigenfunctions and Galerkin method to two‐phase transport models

In this article, we consider a nonlinear partial differential system describing two‐phase transports and try to recover the source term and the nonlinear diffusion term when the state variable is known at different profile times. To this end, we use a POD‐Galerkin procedure in which the proper orthogonal decomposition technique is applied to the ensemble of solutions to derive empirical eigenfunctions. These empirical eigenfunctions are then used as basis functions within a Galerkin method to transform the partial differential equation into a set of ordinary differential equations. Finally, the validation of the used method has been evaluated by some numerical examples. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 456–474, 2007     

New explicit approximate solution of MHD viscoelastic boundary layer flow over stretching sheet

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non‐Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non‐uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.     

Variational iteration method for solving three‐dimensional Navier–Stokes equations of flow between two stretchable disks

The similarity transform for the steady three‐dimensional Navier–Stokes equations of flow between two stretchable disks gives a system of nonlinear ordinary differential equations. In this article, the variational iteration method was used for solving these equations. The results have been compared with the numerical results. This article depicts that the VIM is an efficient and powerful method for solving nonlinear differential equations. This method is applicable to strongly and weakly nonlinear problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A second‐order finite difference approximation for a mathematical model of erythropoiesis

We present a second‐order finite difference scheme for approximating solutions of a mathematical model of erythropoiesis, which consists of two nonlinear partial differential equations and one nonlinear ordinary differential equation. We show that the scheme achieves second‐order accuracy for smooth solutions. We compare this scheme to a previously developed first‐order method and show that the first order method requires significantly more computational time to provide solutions with similar accuracy. We also compare this numerical scheme with other well‐known second‐order methods and show that it has better capability in approximating discontinuous solutions. Finally, we present an application to recovery after blood loss. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Investigation of a powerful analytical method into natural convection boundary layer flow

The natural convection boundary layer flow modeled by a system of nonlinear differential equations is considered. By means of similarity transformation, the non-linear partial differential equations are reduced to a system of two coupled ordinary differential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). Convergence of the obtained series solution is discussed. Finally some figures are illustrated to show the accuracy of the applied method and assessment of various prandtl numbers on the temperature and the velocity is undertaken.     

Meromorphic solutions of nonlinear ordinary differential equations

Exact solutions of some popular nonlinear ordinary differential equations are analyzed taking their Laurent series into account. Using the Laurent series for solutions of nonlinear ordinary differential equations we discuss the nature of many methods for finding exact solutions. We show that most of these methods are conceptually identical to one another and they allow us to have only the same solutions of nonlinear ordinary differential equations.     

Exact solutions of some systems of fractional differential‐difference equations

In this paper, the ‐expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential‐difference equations with the modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential‐difference equation into its differential‐difference equation of integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time‐fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Efficient hybrid method for solving special type of nonlinear partial differential equations

In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh–coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh–coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two‐ and three‐dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed.     

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.     

Application of group analysis to classification of systems of three second‐order ordinary differential equations

Here, we give a complete group classification of the general case of linear systems of three second‐order ordinary differential equations excluding the case of systems which are studied in the literature. This is given as the initial step in the study of nonlinear systems of three second‐order ordinary differential equations. In addition, the complete group classification of a system of three linear second‐order ordinary differential equations is carried out. Four cases of linear systems of equations are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparison of approximate symmetry and approximate homotopy symmetry to the Cahn–Hilliard equation

Complete infinite order approximate symmetry and approximate homotopy symmetry classifications of the Cahn–Hilliard equation are performed and the reductions are constructed by an optimal system of one-dimensional subalgebras. Zero order similarity reduced equations are nonlinear ordinary differential equations while higher order similarity solutions can be obtained by solving linear variable coefficient ordinary differential equations. The relationship between two methods for different order are studied and the results show that the approximate homotopy symmetry method is more effective to control the convergence of series solutions than the approximate symmetry one.     

Positive solutions for singular systems of multi‐point boundary value problems

We study the existence of positive solutions for systems of singular nonlinear second‐order ordinary differential equations subject to multi‐point boundary conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations

The differential transform method is one of the approximate methods which can be easily applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. In this paper, we present the definition and operation of the one-dimensional differential transform and investigate the particular exact solutions of system of ordinary differential equations that usually arise in mathematical biology by a one-dimensional differential transform method. The numerical results of the present method are presented and compared with the exact solutions that are calculated by the Laplace transform method.