A note on the modified simple equation method applied to Sharma-Tasso-Olver equation

Jawad et al. have applied the modified simple equation method to find the exact solutions of the nonlinear Fitzhugh-Naguma equation and the nonlinear Sharma-Tasso-Olver equation. The analysis of the Sharma-Tasso-Olver equation obtained by Jawad et al. is based on variant of the modified simple equation method. In this paper, we provide its direct application and obtain new 1- soliton solutions.     

A note on the number of nodal solutions of an elliptic equation with symmetry

We consider the semilinear problem $-\Delta u+\lambda u={\left|u\right|}^{p-2}u$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega \subset {\mathbb{R}}^N$ is a bounded smooth domain and $2<p<2^1=2N/(N-2)$. We show that if $\Omega$ is invariant under a nontrivial orthogonal involution then, for $\lambda >0$ sufficiently large, the equivariant topology of $\Omega$ is related to the number of solutions which change sign exactly once.     

A full multigrid method for semilinear elliptic equation

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.     

A note on the Fermat equation

Fix one of the base variables x₁, x₂, x₃ in the equation x₁^x4 + x₂^x4 = x₃^x4 subject to x₁, x₂, x₃, x₄ ∈ $\text{Z}$, x₁x₂x₃ ≠ 0, x₄ ≥ 3, gcd(x₁, x₂, x₃) = 1. Then the solutions are bounded. There are no solutions (x₁, x₂, x₃, x₄) with $\text{P(x}{}_{\mathrm{i}}\text{) < ε}{}_0\text{(}\text{log log}\text{|x}{}_{\mathrm{i}}\text{|)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$ for some i ∈ {1, 2, 3}.     

A note on the Goormaghtigh equation

Periodica Mathematica Hungarica - Let p be an odd prime. By using a lower bound for linear forms in logarithms of two algebraic numbers, we prove that if $$p&gt;10^{24}$$ , 2 is a primitive...     

A note on the Riccati differential equation

In this note, we obtain some results for the Riccati differential equations u′=A(z)+u² with nonentire meromorphic functions A(z). Some examples are given to illustrate our some results are sharp.     

A note on the D'Alembert functional equation

It is proved that the analog of DAlemberts equation for a group has symmetric solutions which are central functions only in the commutative case. Necessary and sufficient conditions are given for the representability of the solutions in the form of a semisum of well-defined homomorphisms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 182–183, 1974.     

Multigrid method for coupled semilinear elliptic equation

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.     

A note on new exact solutions for the Kawahara equation using Exp-function method

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, New Exact solutions for the Kawahara equation using Exp-function method, J. Comput. Appl. Math. 233 (2009) 97-102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong.     

A note on skewcirculant preconditioners for elliptic problems

In a recent paper Chan and Chan study the use of circulant preconditioners for the solution of elliptic problems. They prove that circulant preconditioners can be chosen so that the condition number of the preconditioned system can be reduced fromO(n ² ) toO(n). In addition, using the Fast Fourier Transform, the computation of the preconditioner is highly parallelizable. To obtain their result, Chan and Chan introduce a shift /p/n ² for some >0. The aim of this paper is to consider skewcirculant preconditioners, and to show that in this case the condition number ofO(n) can easily be shown without using the somewhat unsatisfactory shift /p/n ². Furthermore, our estimates are more precise.     

A note on partial differential-difference equation

In this paper, we studied a certain partial differential-difference equation which arose from crystal precipitation modeling. The uniqu existence and asymptotic behaviors of the solution were shown     

A probabilistic note on the Cauchy functional equation

Aequationes mathematicae - In this short note we show that a weak version of Bernstein’s characterization of the normal distribution implies the local integrability of a measurable solution...     

Finite element method for a nonsmooth elliptic equation

In this paper, we study the finite element method for a non-smooth elliptic equation. Error analysis is presented, including a priori and a posteriori error estimates as well as superconvergence analysis. We also propose two algorithms for solving the underlying equation. Numerical experiments are employed to confirm our error estimations and the efficiency of our algorithms.     

A note on initial-value formulations of elliptic boundary-value problems

Summary An elliptic boundary-value problem may be formulated as an initial-value problem by approximating it as a two-point boundary-value problem. A method proposed in [6] not requiring this approximation is disproved by means of an example.

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Zusammenfassung Ein elliptisches Randwertproblem lässt sich als Anfangswerproblem, formulieren, indem es durch ein Zweipunkt-Randwertproblem approximiert wird. Eine in [6] vorgeschlagene Methode, die auf diese Approximation verzichtet, wird durchein Beispiel widerlegt.

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The author is thankful for support by the AFOSR, through contract No. F4462-69-C-0124 and NASA-UAH Agreement Modification 4.