Laplacian Boundary Value Problems Research Articles

Positive solutions of singular p-Laplacian BVPs with sign changing nonlinearity on time scales

We investigate a class of singular m -point p -Laplacian boundary value problem on time scales with the sign changing nonlinearity. By using the well-known Schauder fixed point theorem and upper and lower solutions method, some new existence criteria for positive solutions of the boundary value problem are presented. These results are new even for the corresponding differential ( T = R ) and difference equations ( T = Z ) , as well as general time scales setting. As an application, an example is given to illustrate the results.

Some new results on the existence of positive solutions for the one-dimensional [formula omitted]-Laplacian boundary value problems on time scales

This paper is devoted to using the Leggett–Williams fixed point theorem to derive some sufficient conditions for the existence of triple positive solutions to a type of p -Laplacian boundary value problems on time scales. The interesting point is that the nonlinear term f depends on the first-order delta derivative explicitly.

Existence and iteration of positive solutions to a class of Sturm–Liouville-like [formula omitted]-Laplacian boundary value problems

This paper deals with the existence and iteration of positive solutions for the following p -Laplacian differential equation: ( ϕ p ( u ′ ( t ) ) ) ′ + q ( t ) f ( t , u ( t ) , u ′ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , subject to the following Sturm–Liouville-like four-point boundary condition: u ( 0 ) − α u ′ ( ξ ) = 0 , u ( 1 ) + β u ′ ( η ) = 0 where ξ , η ∈ ( 0 , 1 ) . By applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions, we not only obtain the existence of positive solutions for the problems, but also establish iterative schemes for approximating the solutions.

Existence and iteration of positive pseudo-symmetric solutions for a three-point second-order [formula omitted]-Laplacian BVP

In this work, we apply a monotone iterative procedure to prove the existence of positive pseudo-symmetric solutions for a three-point second-order p -Laplacian boundary value problem.

New fixed point theorems and applications to boundary value problems with [formula omitted]-Laplacian

We obtain a new fixed point theorem in cone, which extend the Krasnosel’skii’s compression–expansion theorem in cones. Under a quite relaxed condition two theorems for the existence of positive solutions of p -Laplacian boundary value problems are proved.

A high accurate composite grid method for solving Laplace's boundary value problems with singularities

A high accurate composite grid method for solving Laplace's boundary value problems with singularities

Positive solutions for fourth-order singular p -Laplacian boundary value problems

This article investigates fourth-order singular p-Laplacian boundary value problems (BVPs), and obtains the necessary and sufficient conditions for existence of positive solutions for fourth-order singular p-Laplacian BVPs on closed interval.

Existence and iteration of positive solutions for some [formula omitted]-Laplacian boundary value problems

This paper deals with the existence and iteration of positive solutions for the following p -Laplacian differential equation, ( ϕ p ( u ′ ( t ) ) ) ′ + q ( t ) f ( t , u ( t ) , u ′ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , subject to one of the following boundary conditions, α ϕ p ( u ( 0 ) ) − β ϕ p ( u ′ ( 0 ) ) = 0 , γ ϕ p ( u ( 1 ) ) + δ ϕ p ( u ′ ( 1 ) ) = 0 , or u ( 0 ) − g 1 ( u ′ ( 0 ) ) = 0 , u ( 1 ) + g 2 ( u ′ ( 1 ) ) = 0 . The main tool is the monotone iterative technique. We obtain not only the existence of positive solutions for the problems, but also establish iterative schemes for approximating the solutions.

The High Accurate Block-Grid Method for Solving Laplace's Boundary Value Problem with Singularities

A high accurate difference-analytical method is introduced for the solution of the mixed boundary value problem for Laplace's equation on graduated polygons. The polygon can have broken sections and be multiply connected. The uniform estimate of the error of the approximate solution is of order O(h6 ), whereas it is of order $O(h^{6}/r_{j}^{p-\lambda _{j}})$ for the errors of p-order derivatives (p=1,2,. . .) in a finite neighborhood of reentry vertices; here, h is the mesh step, rj is the distance from the current point to the vertex in question, $\lambda _{j}=1/(a\alpha _{j})$, and a=1 or 2 depending on the types of boundary conditions. Further, $\alpha _{j}\pi $ is the value of the interior angle at the considered vertex. Numerical experiments are illustrated in section \ref{sec8} to support the analysis made.

Uniqueness of positive solutions for generalized Laplacian boundary value problems

In this paper, we establish the uniqueness of positive solutions of generalized Laplacian boundary value problems ▪ where α i , β i ≥ 0 and α i 2 + β i 2 ≠ 0 ( i = 1, 2).

Numerical conformal mapping onto a rectangle with applications to the solution of Laplacian problems

Let F be the function which maps conformally a simply-connected domain Ω onto a rectangle R, so that four specified points on ∂Ω are mapped respectively onto the four vertices of R. In this paper we consider the problem of approximating the conformal map F, and present a survey of the available numerical methods. We also illustrate the practical significance of the conformal map, by presenting a number of applications involving the solution of Laplacian boundary value problems.

A mixed boundary value problem for the Laplacian

On etudie le probleme de Zaremba concernant le laplacien sur des domaines planaires simplement connexes

A nonconforming combined method for solving Laplace's boundary value problems with singularities

A nonconforming combined method for solving Laplace's boundary value problems with singularities

A nonconforming combined method for solving Laplace's boundary value problems with singularities

For solving Laplace's boundary value problems with singularities, a nonconforming combined approach of the Ritz-Galerkin method and the finite element method is presented. In this approach, singular functions are chosen to be admissible functions in the part of a solution domain where there exist singularities; and piecewise linear functions are chosen to be admissible functions in the rest of the solution domain. In addition, the admissible functions used here are constrained to be continuous only at the element nodes on the common boundary of both methods. This method is nonconforming; however, the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used.

AN ASYMPTOTIC INTEGRAL EQUATION METHOD FOR THE STUDY OF HEAT TRANSFER IN LONG, THIN DOMAINS

A numerical method is presented for the computationally efficient and accurate solution of heat transfer boundary-value problems (BVPs) within long, slender bodies (e.g., fin assemblies). The method is based on the well-known boundary integral equation (BIE) technique for the solution of steady-state two-dimensional Laplacian BVPs. Whereas the classical BIE is satisfactory for BVPs in which the solution domain has a low length-to-width aspect ratio, it is established that the accuracy and efficiency of the classical BIE deteriorate rapidly for aspect ratios of order greater than O(10). In the present paper the BIE is modified to cope with BVPs in which aspect ratios of O(1000) are encountered. This enables accurate treatment of more physically viable fin assembly geometries that were hitherto beyond the scope of the BIE. The modification essentially incorporates the asymptotic (analytically derived) solution in the majority of the fin, leaving the BIE to solve the BVP numerically in only a minor part of t...

An Asymptotic Integral Equation Method for the Study of Heat Transfer in Long, Thin Domains

A numerical method is presented for the computationally efficient and accurate solution of heat transfer boundary-value problems (BVPs) within long, slender bodies (e.g., fin assemblies). The method is based on the well-known boundary integral equation (BIE) technique for the solution of steady-state two-dimensional Laplacian BVPs. Whereas the classical BIE is satisfactory for BVPs in which the solution domain has a low length-to-width aspect ratio, it is established that the accuracy and efficiency of the classical BIE deteriorate rapidly for aspect ratios of order greater than O(10). In the present paper the BIE is modified to cope with BVPs in which aspect ratios of O(1000) are encountered. This enables accurate treatment of more physically viable fin assembly geometries that were hitherto beyond the scope of the BIE. The modification essentially incorporates the asymptotic (analytically derived) solution in the majority of the fin, leaving the BIE to solve the BVP numerically in only a minor part of t...

Difference methods on a digital computer for laplacian boundary value and eigenvalue problems

Difference methods on a digital computer for laplacian boundary value and eigenvalue problems

Coding of a Laplace boundary value problem for the UNIVAC

Coding of a Laplace boundary value problem for the UNIVAC

Coding of a Laplace Boundary Value Problem for the UNIVAC

Coding of a Laplace Boundary Value Problem for the UNIVAC