On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation

The modified method of simplest equation is powerful tool for obtaining exact and approximate solutions of nonlinear PDEs. These solutions are constructed on the basis of solutions of more simple equations called simplest equations. In this paper we study the role of the simplest equation for the application of the modified method of simplest equation. We follow the idea that each function constructed as polynomial of a solution of a simplest equation is a solution of a class of nonlinear PDEs. We discuss three simplest equations: the equations of Bernoulli and Riccati and the elliptic equation. The applied algorithm is as follows. First a polynomial function is constructed on the basis of a simplest equation. Then we find nonlinear ODEs that have the constructed function as a particular solution. Finally we obtain nonlinear PDEs that by means of the traveling-wave ansatz can be reduced to the above ODEs. By means of this algorithm we make a first step towards identification of the above-mentioned classes of nonlinear PDEs.     

Asymptotic expansions for degenerate parabolic equations

We prove asymptotic convergence results for some analytical expansions of solutions to degenerate PDEs with applications to financial mathematics. In particular, we combine short-time and global-in-space error estimates, previously obtained in the uniformly parabolic case, with some a priori bounds on “short cylinders”, and we achieve short-time asymptotic convergence of the approximate solution in the degenerate parabolic case.     

On error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains

Numerical verification methods, so-called Nakao's methods, on existence or uniqueness of solutions to PDEs have been developed by Nakao and his group including the authors. They are based on the error estimation of approximate solutions which are mainly computed by FEM.     

The Convergence Analysis of the Decomposition Method for the (1+1)-Parabolic Problem in Nonuniform Media

A modified Adomian decomposition method is used to find approximate solutions for the reaction diffusion convection equation in nonuniform medium, i.e., with space-dependent coefficients. In this approach, the solutions are found in the form of a convergent power series with easily computed components. Convergence analysis of modified Adomian series solution for a class of these type of PDEs is discussed.     

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

We derive a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge–Kutta Collocation (RK-C) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ODEs and parabolic PDEs. The key ingredients in deriving these bounds are appropriate one-degree higher continuous reconstructions of the approximate solutions and pointwise error representations. The reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time-step; they do not hinge on asymptotics.     

Analytic approximate solutions to Burgers,Fisher, Huxley equations and two combined forms of these equations

In this paper, we are giving analytic approximate solutions to a class of nonlinear PDEs using the homotopy analysis method (HAM). The Burgers, Fisher, Huxley, Burgers–Fisher and Burgers–Huxley equations are considered. We aim two goals: one is to highlight the efficiency of HAM in solving this class of PDEs and the other is that, although the considered equations have different combinations of nonlinear terms, when applying HAM, we use the same initial guess, the same auxiliary linear operator and the same auxiliary function for all of them.     

On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis–Processi equation and b-equation

The method of simplest equation is powerful tool for obtaining exact and approximate solutions of nonlinear PDEs. Here we extend the class of equations which can be treated by the method in such a way that the classes of equations considered in our previous work are particular cases of the extended class of equations. As examples for application of the methodology we obtain exact traveling-wave solutions of the generalized Degasperis–Processi equation and of the b-equation. As simplest equations we use the equations of Bernoulli and Riccati. We investigate the possibility for obtaining these solutions also by means of the exp-function method. This lead us to propose a generalized version of the exp-function method in Section 5.     

Error analysis and applications of the Fourier-Galerkin Runge-Kutta schemes for high-order stiff PDEs

An integrating factor mixed with Runge-Kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations. In this paper, the novel hybrid Fourier-Galerkin Runge-Kutta scheme, with the aid of an integrating factor, is proposed to solve nonlinear high-order stiff PDEs. Error analysis and properties of the scheme are provided. Application to the approximate solution of the nonlinear stiff Korteweg-de Vries (the 3rd order PDE, dispersive equation), Kuramoto-Sivashinsky (the 4th order PDE, dissipative equation) and Kawahara (the 5th order PDE) equations are presented. Comparisons are made between this proposed scheme and the competing method given by Kassam and Trefethen. It is found that for KdV, KS and Kawahara equations, the proposed method is the best.     

Exact and analytical solutions for a nonlinear sigma model

We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error‐minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximate Analysis of a Nonlinear Dissipative Model

From the approximate symmetry point of view, a perturbed system of partial differential equations (PDEs) for viscoelastic media is investigated. The corresponding complete approximate symmetry classification is derived and a fundamental task of the paper is the proof of a theorem that explores the relationship between the symmetries of our model and other models. In some physical cases, approximate solutions are computed by means of the approximate generator of the first order approximate group of transformations.     

一类带强色散项DGH方程的精确解

利用改进的F-展开法,求出了一类带强色散项DGH方程的一系列类孤子解,三角函数周期解和有理数解,方程结合了KdV方程的线性色散项和C-H方程的非线性色散项.而且改进的F-展开法在借助于计算机符号系统Mathematica(Maple)下,操作方便,适用于大量的非线性偏微分方程(组),并有助于发现新解.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方 法,即改进的广义射影Ricccati方程方法,求解(2 1)维色散长波方程,得到该方程的新的 更一般形式的行波解,包括扭状孤波解,钟状解,孤子解和周期解．并对部分新形式孤波解画 图示意．     

Periodic wave solutions of coupled integrable dispersionless equations by residue harmonic balance

We introduce the residue harmonic balance method to generate periodic solutions for nonlinear evolution equations. A PDE is firstly transformed into an associated ODE by a wave transformation. The higher-order approximations to the angular frequency and periodic solution of the ODE are obtained analytically. To improve the accuracy of approximate solutions, the unbalanced residues appearing in harmonic balance procedure are iteratively considered by introducing an order parameter to keep track of the various orders of approximations and by solving linear equations. Finally, the periodic solutions of PDEs result. The proposed method has the advantage that the periodic solutions are represented by Fourier functions rather than the sophisticated implicit functions as appearing in most methods.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方法,即改进的广义射影Ricccati方程方法, 求解(2+1)维色散长波方程, 得到该方程的新的更一般形式的行波解, 包括扭状孤波解, 钟状解,孤子解和周期解. 并对部分新形式孤波解画图示意.     

Geometry of PDEs: I: Integral bordism groups in PDEs

We improve some our previous theorems on the calculation of integral bordism groups of formally integrable and completely integrable PDEs, emphasizing the role played by singular solutions and weak solutions. Some applications to interesting PDEs, defined on finite dimensional manifolds, are also considered.     

Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation

In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.     

Kinetic decomposition for singularly perturbed higher order partial differential equations

In this paper, we consider singularly perturbed higher order partial differential equations. We establish the condition under which the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed in Hwang and Tzavaras (Comm. Partial Differential Equations 27 (2002) 1229). First, we obtain the approximate transport equation for the given dispersive equations. Then using the averaging lemma, we obtain the convergence.     

Invariants of multi-parameter approximate transformation groups

The problem of finding all functionally independent invariants for a given multi-parameter approximate transformation group is considered. The criterion for approximate function to be invariant under an approximate transformation group is similar to that of Lie's theory and is based on operators of the corresponding approximate Lie algebra. The calculation of invariants of an approximate transformation group reduces to solving the system of linear nonhomogeneous first-order PDEs. The sufficient conditions of compatibility and completeness for these systems are proved.     

Unifying the steady state resonant solutions of the periodically forced KdVB, mKdVB, and eKdVB equations

The periodically forced extended KdVB (eKdVB) equation, which contains both KdVB and modified KdVB (mKdVB) equations as special cases, is known to possess a rich array of resonant steady solutions. We present an analytic methodology based on singular perturbation and asymptotic matching in order to illustrate and approximate these solutions in the limit that the dispersive effects are small relative to the nonlinear and forcing terms. Weak Burgers damping is also included at the same order as dispersion. Solutions across the resonant band may be constructed and show good agreement with solutions of the full equation, showing clearly the role of the various physical effects. In this way, direct comparisons and connections are made between the various classes of KdVB equations, illustrating, in particular, the underlying mathematical connections between the KdVB and mKdVB equations.