Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems

For elliptic boundary value problems of the form ?ΔU+F(x, U, U _x )=0 on Ω,B[U]=0 on ?Ω, with a nonlinearityF growing at most quadratically with respect to the gradientU _x and with a mixed-type linear boundary opeatorB, a numerical method is presented which can be used to prove the existence of a solution within a “close”H _1,4(Ω)-neighborhood of some approximate solution ω∈H ₂(Ω) satisfying the boundary condition, provided that the defect-norm ∥?Δω +F(·, ω, ω _x )∥₂ is sufficiently small and, moreover, the linearization of the given problem at ω leads to an invertible operatorL. The main tools are explicit Sobolev imbeddings and eigenvalue bounds forL or forL*L. All kinds of monotonicity or inverse-positivity assumptions are avoided.     

Sinc-Galerkin method for solving nonlinear weakly singular two point boundary value problems

A study of Sinc-Galerkin method based on double exponential transformation for solving a class of nonlinear weakly singular two point boundary value problems with non-homogeneous boundary conditions is given. The properties of the Sinc-Galerkin approach are utilized to reduce the computation of nonlinear problem to nonlinear system of equations with unknown coefficients. This method tested on several test examples. We compare our numerical results with several numerical results of existing methods. The demonstrated results confirm that proposed method is considerably efficient, accurate nature and rapidly converge.     

A survey on various computational techniques for nonlinear elliptic boundary value problems

In this paper a classification and a survey on numerical techniques for solving nonlinear (quasilinear, semilinear, superlinear, sublinear) elliptic boundary value problems between 2001 and 2006 have been presented and discussed the nature of positive solution of the various problems. The introduction of the methods and results presented by different researchers are summarized.     

Existence of positive solutions of singular elliptic boundary value problems in a ball

This paper deals with existence theorems of positive solutions for singular elliptic boundary value problems in a ball. The results of this paper are generalizations of those proved by authors cited in the references.     

Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems

Cui et al. [M. Cui and F. Geng, Solving singular two point boundary value problems in reproducing kernel space, J. Comput. Appl. Math. 205 (2007), pp. 6–15; H. Yao and M. Cui, A new algorithm for a class of singular boundary value problems, Appl. Math. Comput. 186 (2007), pp. 1183–1191] presents an algorithm to solve a class of singular linear boundary value problems in the reproducing kernel space. In this paper, we will present three new algorithms to solve a class of singular weakly nonlinear boundary value problems in reproducing kernel space. The algorithms are efficiently applied to solving some model problems. It is demonstrated by the numerical examples that those algorithms are highly accurate.     

A new method for automatical computation of error bounds for the set of all solutions of nonlinear boundary value problems

We consider nonlinear boundary value problems with arbitrarily many solutionsuεC ² [a, b]. In this paper an Algorithm will be established for a priori bounds \(\bar u,\bar d \in C[a,b]\) with the following properties:

1. For every solutionu of the nonlinear problem we obtain $$\bar u(x) \leqslant u(x) \leqslant \bar u(x), - \bar d(x) \leqslant u'(x) \leqslant \bar d(x)$$ for any,xε[a, b].
2. The bounds \(\bar u\) and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaara% aaaa!36EE!\[\bar d\] are defined by the use of the functions exp, sin and cos.
3. We use neither the knowledge of solutions nor the number of solutions.

     

A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems

In this paper, we propose an algorithm for solving the nonlinear two-point boundary value problem

that has at least one positive solution [1–6] for λ in a compatible interval. Our method stems mainly from combining the decomposition series solution obtained by Adomian decomposition method with Padé approximates. The validity of the approach is verified through illustrative numerical examples     

Solving elliptic boundary value problems by double sweep method

In this paper we give the conditions of applying the double sweep method to certain boundary value problems of elliptic type. We also extend the results obtained in a previous paper (double sweep by jumps) and we remark the possibility to use the Cooley and Tukey's algorithm (FFT).     

Numerical computations for singular semilinear elliptic boundary value problems

The paper studies a class of Dirichlet problems with homogeneous boundary conditions for singular semilinear elliptic equations in a bounded smooth domain in

. A numerical method is devised to construct an approximate Green's function by using radial basis functions and the method of fundamental solutions. An estimate of the error involved is also given. A weak solution of the above given problem is a solution of its corresponding nonlinear integral equation. A computational method is given to find the minimal weak solution U, and the critical index λ* (such that a weak solution U exists for λ < λ*, and U does not exist for λ > λ*).     

Spectral projective newton-methods for quasilinear elliptic boundary value problems

We consider Newton-like methods for the solution of quasilinear elliptic boundary value problems. The quasilinear problems are linearized by a Newton-method and the linear problems are approximately solved by a spectral projection method (e.g., the Ritz-Galerkin or the collocation method). convergence results are derived that show the spectral accuracy of this method. The results are of a local type which means that we assume the starting approximation to be sufficiently near to the exact solution.     

Domain decomposition method for elliptic mixed boundary value problems

A method for solving the elliptic second order mixed boundary value problem is discussed. The finite element problem is divided into subproblems, associated with subregions into which the region has been partitioned, and an auxiliary problem connected with intersect curves. The subproblems are solved directly, while the auxiliary problem is handled by a conjugate gradient method. The rate of the convergence of the cg-method is discussed also for the cases when the Neumann and Dirichlet boundary conditions change at points belonging to the intersecting curves. Results from numerical experiments are also reported.     

Existence and multiplicity results for nonlinear boundary value problems

This paper studies the existence of positive solutions for a class of boundary value problems of elliptic degenerate equations. By using bifurcation and fixed point index theories in the frame of approximation arguments, the criteria of the existence, multiplicity and nonexistence of positive solutions are established.     

An adaptive,multi-level method for elliptic boundary value problems

Subroutine PLTMG is a Fortran program for solving self-adjoint elliptic boundary value problems in general regions ofR ². It is based on a piecewise linear triangle finite element method, an adaptive grid refinement procedure, and a multi-level iterative method to solve the resulting sets of linear equations. In this work we describe the method and present some numerical results and comparisons.     

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi’s elliptic functions.For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi’s sn or cn functions. Examples illustrate key steps of the algorithms.The new algorithms are implemented in Mathematica. The package PDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed.A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.     

Partial evaluation of the discrete solution of elliptic boundary value problems

The technique of hierarchical matrices is used to construct a solution operator for a discrete elliptic boundary value problem. The solution operator can be determined once for all from a recursive domain decomposition structure. Then, given boundary values and a source term, the solution can be evaluated by applying the solution operator. The complete procedure yields all components of the solution vector. The data size and computational cost is $O(n\hbox {log}^{*}n),$ where $n$ is the number of unknowns. Once the data of the solution operator are constructed, components related to small subdomains can be truncated. This reduces the storage amount and still enables a partial evaluation of the solution (restricted to the skeletons of the remaining subdomains). The latter approach is in particular suited for problems with oscillatory coefficients, where one is not interested in all details of the solution.