Laplacian Boundary Value Problems Research Articles

On singular φ−Laplacian BVPs of nonlinear fractional differential equation

This paper investigates the existence of multiple positive solutions for a class of φ−Laplacian boundary value problem with a nonlinear fractional differential equation and fractional boundary conditions. Multiple solutions are proved under slight conditions on a possibly degenerating source term. Approximation techniques together with the fixed-point index theory a on cone of a Banach space are employed. Some illustrating examples of are also supplied. Mathematics Subject Classification (2010): 34A08, 34B15, 34B18, 47H10. Received 27 July; Accepted 13 December 2021

Positive solutions for a fourth-order p-Laplacian boundary value problem

Abstract The authors consider a fourth-order boundary value problem with a p-Laplacian appearing in both the equation and the boundary conditions. Upper and lower estimates for positive solutions to the problem are proved. Sufficient conditions for the existence and nonexistence of positive solutions to the problem are obtained and illustrated with an example.

Existence of Nontrivial Weak Solutions for Some Discrete p(.)-Laplacian Boundary Value Problems in n-Dimensional

Existence of Nontrivial Weak Solutions for Some Discrete p(.)-Laplacian Boundary Value Problems in n-Dimensional

Uniqueness results for a mixed $ p $-Laplacian boundary value problem involving fractional derivatives and integrals with respect to a power function

<abstract><p>This paper is concerned with a mixed $ p $-Laplacian boundary value problem involving right-sided and left-sided fractional derivatives and left-sided integral operators with respect to a power function. We prove the uniqueness of positive solutions for the given problem for the cases $ 1 < p \le 2 $ and $ p > 2 $ by applying an efficient novel approach together with the Banach contraction mapping principle. Estimates for Green's functions appearing in the solution of the problem at hand are also presented. Examples are given to illustrate the obtained results.</p></abstract>

Positive solutions to mixed fractional p-Laplacian boundary value problems

Abstract In this paper, we discuss the existence and uniqueness of a positive solution for a p-Laplacian differential equation containing left and right Caputo derivatives. By the help of the Guo–Krasnoselskii theorem, we prove the existence of at least one positive solution. The existence of a unique positive solution is established under the assumption that the corresponding operator is α-concave and increasing. Numerical examples are given to check the obtained results.

Existence of solutions for weighted p(t)-Laplacian mixed Caputo fractional differential equations at resonance

Using Mawhin?s coincidence degree theory, we investigate the existence of solutions for a class of weighted p(t)-Laplacian boundary value problems at resonance and involving left and right Caputo fractional derivatives. An example is provided to illustrate the main existence results.

On solvability of some $ p $-Laplacian boundary value problems with Caputo fractional derivative

<abstract><p>The solvability of some $ p $-Laplace boundary value problems with Caputo fractional derivative are discussed. By using the fixed-point theory and analysis techniques, some existence results of one or three non-negative solutions are obtained. Two examples showed that the conditions used in this paper are somewhat easy to check.</p></abstract>

Steklov Expansion Method for Regularized Harmonic Boundary Value Problems

A new type of meshless method is proposed in this paper to solve regularized Laplacian boundary value problems of the form In this method, solving the Laplacian boundary value problem is starting from finding regularized Steklov eigenpairs which could provide the orthonormal basis of the space of solutions. The solutions are represented by orthogonal regularized Steklov eigenfunctions. Error bounds of Steklov approximations are provided in terms of the traces spaces of solutions. When rectangular domains are considered, explicit formulas for the regularized Steklov eigenpairs are provided. Some numerical experiments are presented to support the efficiency and accuracy of the new method.

Steklov approximations of Green’s functions for Laplace equations

Purpose This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator. Design/methodology/approach The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary. Findings The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary. Originality/value This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.

On p–Laplacian boundary value problems involving Caputo–Katugampula fractional derivatives

In this paper, we study the existence and uniqueness of solutions for a p–Laplacian boundary value problem defined by semilinear fractional system that involves Caputo–Katugampola fractional derivatives. Our main results rely on the implementation of the Banach and Schauder fixed point theorems. An example is introduced to expose the applicability of the theoretical findings.

Existence and Multiplicity Results for Nonlocal Boundary Value Problems with Strong Singularity

In this paper, we study singular φ -Laplacian nonlocal boundary value problems with a nonlinearity which does not satisfy the L 1 -Carathéodory condition. The existence, nonexistence and/or multiplicity results of positive solutions are established under two different asymptotic behaviors of the nonlinearity at ∞.

Existence of Positive Solutions to Singular φ-Laplacian Nonlocal Boundary Value Problems when φ is a Sup-multiplicative-like Function

In this paper, using a fixed point index theorem on a cone, we present some existence results for one or multiple positive solutions to φ -Laplacian nonlocal boundary value problems when φ is a sup-multiplicative-like function and the nonlinearity may not satisfy the L 1 -Carath e ´ odory condition.

FEM and BEM Implementations of a High Order Surface Impedance Boundary Condition for Three-Dimensional Eddy Current Problems

We implement a three-dimensional formulation for eddy current problems based on the reduced magnetic scalar potential enforcing a high order surface impedance boundary condition (SIBC) which takes into account the curvatures of the conductor surface. Based on perturbation theory, the formulation reduces to three Laplace boundary value problems with Neumann boundary conditions and hence can be easily implemented in any existing Finite Element Method (FEM) or Boundary Element Method (BEM) code for the Laplace equation. The appropriate choice of the small parameter in the perturbation approach correctly represents the order of accuracy of the SIBC. The validation is carried out by comparison with full FEM solutions of a canonical test problem and of a more realistic example of a non-destructive testing probe. The validity of the extension of a high order SIBC to lower frequencies is verified and the fields can be obtained at any frequency in the range of interest once the formulation is solved only once.

Existence results for a class of p-q Laplacian semipositone boundary value problems

Existence results for a class of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> Laplacian semipositone boundary value problems

Degenerate Laplacian describing topologically constrained diffusion: helicity constraint as an alternative to ellipticity

We study a second-order non-elliptic partial differential operator that generalizes the classical Laplacian with a degenerate coefficient matrix. The theory is motivated by the statistical mechanics of topologically constrained particles. In the context of diffusion models, the coefficient matrix is given by , with a generalized Poisson matrix that dictates particle dynamics. When is the symplectic matrix of canonical Hamiltonian systems, the operator reduces to the classical Laplacian. However, topological constraints make the matrix degenerate. The ellipticity is broken in the direction parallel to the nullity of . We call such a diffusion operator an orthogonal Laplacian and denote it by . When the nullity foliates the space, reduces into a classical Laplacian on the leaf that is orthogonal to the null space. However, when the nullity of is not integrable (which is the case when has helicity), is akin to elliptic operators having some useful properties of convex analysis. We show that defines an inner product of a Sobolev-like Hilbert space, and is a coercive functional in the L2 topology. Constructing a trace theorem, we discuss a boundary value problem (the orthogonal Poisson equation) for the orthogonal Laplacian, and obtain a unique weak solution by application of Riesz’s representation theorem.

Solutions to nonlinear second-order three-point boundary value problems of dynamic equations on time scales

In this paper, we consider existence criteria of three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales. To show our main results, we apply the well-known Leggett-Williams fixed point theorem. Moreover, we present some results for the existence of single and multiple positive solutions for boundary value problems on time scales, by applying fixed point theorems in cones. The conditions we used in the paper are different from those in [Dogan A. On the existence of positive solutions for the one-dimensional $ p $-Laplacian boundary value problems on time scales. Dynam Syst Appl 2015; 24: 295-304].

Extremal solutions for p‐Laplacian boundary value problems with the right‐handed Riemann‐Liouville fractional derivative

The paper is concerned with the solvability for several nonlinear boundary value problems of fractional p‐Laplacian differential equation involving the right‐handed Riemann‐Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, sufficient conditions for existence and uniqueness of extremal solutions are obtained and they extend existing results. At last, two examples are provided to illustrate the results.

Existence of positive solutions for systems of nonlinear fractional differential equation with p-Laplacian

In this paper, we establish sufficient conditions for the existence of positive solutions for a system of nonlinear fractional [Formula: see text]-Laplacian boundary value problems under different combinations of superlinearity and sublinearity of the nonlinearities via the Guo–Krasnosel’skii fixed point theorem. Moreover, an example is given to illustrate our results.

A posteriori error estimates for the generalized overlapping domain decomposition method for a parabolic variational equation with mixed boundary condition

The paper deals with a posteriori error estimates for the generalized overlapping domain decomposition method with mixed boundary condition the interfaces for parabolic variational equation with Laplace boundary value problems are proved using by the theta time scheme combined with Galerkin spatial approximation.

Linearized stability implies dynamic stability for equilibria of 1-dimensional, p-Laplacian boundary value problems

AbstractWe consider the parabolic, initial-boundary value problem 1$$\matrix{ {\displaystyle{{\partial v} \over {\partial t}} = \Delta _p(v) + f(x,v),} &amp; {{\rm in}({\rm - 1},{\rm 1}) \times ({\rm 0},\infty ),} \cr {v( \pm 1,t) = 0,} \hfill \hfill \hfill &amp; {{\rm t}\in [{\rm 0},\infty ),} \hfill \hfill \cr {v = v_0\in C_0^0 ([-1,1]),} &amp; {{\rm in}[{\rm - 1},{\rm 1}] \times \{ {\rm 0}\} ,} \cr } $$ where Δp denotes the p-Laplacian on ( − 1, 1), with p &gt; 1, and the function f:[ − 1, 1] × ℝ → ℝ is continuous, and the partial derivative fv exists and is continuous and bounded on [ − 1, 1] × ℝ. It will be shown that (under certain additional hypotheses) the ‘principle of linearized stability’ holds for equilibrium solutions u0 of (1). That is, the asymptotic stability, or instability, of u0 is determined by the sign of the principal eigenvalue of a suitable linearization of the problem (1) at u0. It is well-known that this principle holds for the semilinear case p = 2 (Δ2 is the linear Laplacian), but has not been shown to hold when p ≠ 2.We also consider a bifurcation type problem similar to (1), having a line of trivial solutions. We characterize the stability or instability of the trivial solutions, and the bifurcating, non-trivial solutions, and show that there is an ‘exchange of stability’ at the bifurcation point, analogous to the well-known result when p = 2.