Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu f(u) =0 in Ω, u=0 on Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u~3,u~2(u-p),u~2(u~2 -p),     

Numerical approximation of solution for the coupled nonlinear Schrödinger equations

In this article, a compact finite difference scheme for the coupled nonlinear Schrödinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ² + h⁴) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.     

A remark on the theory of nonlinear differential equations with the Painlevé property

We study the relationship between the resonance method, which is also referred to as the Painlevé formal test, for nonlinear differential equations resolved for the higher derivative and the representation of these equations in the form of Briot-Bouquet systems.     

A method for the numerical solution of the Painlevé equations

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.     

Dynamical understanding of loop soliton solution for several nonlinear wave equations

It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions for four nonlinear wave equations are discussed. Exact explicit parametric representations of some special travelling wave solutions are given. The results of this paper show that a loop solution consists of three different breaking travelling wave solutions. It is not one real loop soliton travelling wave solution.     

A class of fully nonlinear 2×2 systems of partial differential equations

This paper is a study of certain fully nonlinear 2×2 systems of partial differential equations in one space variable and time. The nonlinearity contains a term proportional to |?U/?x| where U = U(x,t) $isinv; ?² is the unknown function and |·| is the Euclidean norm on ?²; i.e., a term homogeneous of degree 1 in ?U/?x and singular at the origin. Such equations are motivated by hypoplasticity. The paper introduces a notion of hyperbolicity for such equations and, in the hyperbolic case, proves existence of solutions for two initial value problems admitting similarity solutions: the Riemann problem and the scale-invariant problem. Uniqueness is addressed in a companion paper.     

A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann–Liouville derivative

In this work, by means of the fixed point theorem in a cone, we establish the existence result for a positive solution to a kind of boundary value problem for a nonlinear differential equation with a Riemann–Liouville fractional order derivative. An example illustrating our main result is given. Our results extend previous work in the area of boundary value problems of nonlinear fractional differential equations [C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010) 1050–1055].     

Use of the D‐method for differential equations

Harmful effects are pointed out of the use of the D‐method for finding particular integrals of linear differential equations with constant coefficients. These particularly apply for students without much mathematical aptitude, and for those who will go on to study quantum mechanics. An alternative method is outlined.     

Reduced differential transform method for nonlinear integral member of Kadomtsev–Petviashvili hierarchy differential equations

The main objective of this paper is to use the reduced differential to transform method (RDTM) for finding the analytical approximate solutions of two integral members of nonlinear Kadomtsev–Petviashvili (KP) hierarchy equations. Comparing the approximate solutions which obtained by RDTM with the exact solutions to show that the RDTM is quite accurate, reliable and can be applied for many other nonlinear partial differential equations. The RDTM produces a solution with few and easy computation. This method is a simple and efficient method for solving the nonlinear partial differential equations. The analysis shows that our analytical approximate solutions converge very rapidly to the exact solutions.     

Extension of Floquet''''s theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations

We establish conditions for the existence of an invariant set of the system of differential equations

Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm integral equations

A numerical method based on an m-set of general, orthogonal triangular functions (TF) is proposed to approximate the solution of nonlinear Volterra–Fredholm integral equations. The orthogonal triangular functions are utilized as a basis in collocation method to reduce the solution of nonlinear Volterra–Fredholm integral equations to the solution of algebraic equations. Also a theorem is proved for convergence analysis. Some numerical examples illustrate the proposed method.     

Comments on “A one-step optimal homotopy analysis method for nonlinear differential equations”

The above mentioned paper contains some fundamental mistakes and misinterpretations along with a false conclusion. Applying the optimal homotopy asymptotic method (OHAM) in an incorrect manner, Niu and Wang have drawn the false conclusion that this approach is not efficient in practice because it is time-consuming for high-order of approximation. We emphasized the presence of some evident mistakes and misinterpretations in their paper and we proved that OHAM is very efficient in practice since we solved all three examples analyzed by Niu and Wang using only the first-order of approximation, which yields accurate results. We demonstrate that OHAM does not need high-orders of approximation as Niu and Wang suggests and we show that the main strength of OHAM is its rapid convergence, contradicting Niu and Wang’s assumption.     

The computation of consistent initial values for nonlinear index-2 differential–algebraic equations

The computation of consistent initial values for differential–algebraic equations (DAEs) is essential for starting a numerical integration. Based on the tractability index concept a method is proposed to filter those equations of a system of index-2 DAEs, whose differentiation leads to an index reduction. The considered equation class covers Hessenberg-systems and the equations arising from the simulation of electrical networks by means of Modified Nodal Analysis (MNA). The index reduction provides a method for the computation of the consistent initial values. The realized algorithm is described and illustrated by examples.     

Rational Jacobi elliptic solutions for nonlinear differential–difference lattice equations

In this work, we present a direct new method for constructing the rational Jacobi elliptic solutions for nonlinear differential–difference equations, which may be called the rational Jacobi elliptic function method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential–difference equations in mathematical physics via the lattice equation. The proposed method is more effective and powerful for obtaining the exact solutions for nonlinear differential–difference equations.