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In this paper, we introduce a two-point nonlinear boundary value problem for a finite fractional difference equation. An associated Green’s function is constructed as a series of functions and some of its properties are obtained. Some existence results are deduced from fixed point theory and lower and upper solutions Alberto Cabada is partially supported by Xunta de Galicia (Spain), project EM2014/032 and AIE, Spain and FEDER, grant PID2020-113275GB-I00. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature SI

WOS: 000507424500001 In this paper, a numerical approach is suggested to find a semi-analytical solution for the common nonlinear boundary value problems (BVPs) of cantilever-type micro-electromechanical system (MEMS) and nano-electromechanical system (NEMS) with a set of model parameters. The nonlinear BVPs that are studied involve the states of the single and double cantilever-shaped beams under the influence of Casimir and Van der Waals force for proper distances of separation. The method is based upon an integral operator that is formed considering Green's function connected with the execution of Picard's or Mann's fixed point schemes. The numerical results for different cases of beam are presented and compared with those obtained by previous works to show the applicability, efficiency, and high accuracy of the suggested method.

Solutions to nonlinear boundary value problems modelling physical phenomena in engineering applications have traditionally been approximated using numerical methods. More recently, several semi-analytical methods have been developed and used extensively in diverse engineering applications. This work compares the method of variation of parameters to semi-analytical methods for solving nonlinear boundary value problems arising in engineering. The accuracy and efficiency of the method of variation of parameters are compared to those of two widely used semi-analytical methods, the Adomian decomposition method and the differential transformation method, for three practical engineering boundary value problems: (1) the deflection of a cantilevered beam with a concentrated load, (2) an adiabatic tubular chemical reactor which processes an irreversible exothermal reaction, and (3) the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylinder conduit. The accuracy and convergence of each method is investigated using the error remainder function. The method of variation of parameters is significantly more efficient than both the semi-analytical methods and traditional numerical methods while maintaining comparable accuracy. Unlike the semi-analytical methods, the efficiency of variation of parameters is independent of the nonlinearity. Variation of parameters is shown to be an attractive alternative to semi-analytical methods and traditional numerical methods for solving boundary value problems encountered in engineering applications in which solution efficiency is important.

In this paper, we used some theorems of fixed point for studying the results of existence and uniqueness for Hilfer-Hadamard-Type fractional differential equations, \[_{H}D^{\alpha,\beta}x(t)+f(t,x(t))=0, \hbox{ on the interval } J:=(1,e]\] with nonlinear boundary value problems \[x(1+\epsilon)=\sum_{i=1}^{n-2}\nu_{i}x(\zeta_{i}), \quad\quad\quad~_{H}D^{1,1}x(e) = \sum_{i=1}^{n-2} \sigma_i~_{H} D^{1,1}x(\zeta_{i})\], Comment: 17 pages. arXiv admin note: text overlap with arXiv:1801.10400

In the numerical integration of the second-order nonlinear boundary value problem (BVP), the right boundary condition plays the role as a target equation, which is solved either by the half-interval method (HIM) or a new derivative-free Newton method (DFNM) to be presented in the paper. With the help of a boundary shape function, we can transform the BVP to an initial value problem (IVP) for a new variable. The terminal value of the new variable is expressed as a function of the missing initial value of the original variable, which is determined through a few integrations of the IVP to match the target equation. In the new boundary shape function method (NBSFM), we solve the target equation to obtain a highly accurate missing initial value, and then compute a precise solution. The DFNM can find more accurate left boundary values, whose performance is superior than HIM. Apparently, DFNM converges faster than HIM. Then, we modify the Lie-group shooting method and combine it to the BSFM for solving the nonlinear BVP with Robin boundary conditions. Numerical examples are examined, which assure that the proposed methods together with DFNM can successfully solve the nonlinear BVPs with high accuracy.

This paper provides sufficient conditions to guarantee the existence, non-existence and multiplicity of solutions for a third order eigenvalue fully differential equation, coupled with three point boundary value conditions. Although the change of sign, some bounds for the second derivative of the Green's function are obtained, which allow to define a different kind of cone that, as far as we know, has not been previously used in the literature. The main arguments are based on the fixed point index theory for bounded and unbounded sets. Some examples are presented in order to show that the different existence theorems proved are not comparable SI

In this paper, a new technique for solving a class of nonlinear Boundary Value Problems (BVPs) is introduced. Convergence of the series solution obtained from the proposed technique is proved. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. Some numerical examples are introduced to verify the efficiency of the new technique.

For a class of linear partial differential operators of mixed elliptic–hyperbolic type with homogeneous Dirichlet data on the entire boundary of suitable planar domains, we exploit the recent spectral theory of Lupo et al. [2] to establish a Fredholm alternative for weak solutions of the linear Dirichlet problem. This alternative is then used to study nonlinear Dirichlet problems with at most asymptotically linear nonlinearities, both in resonant and nonresonant cases. In particular, we obtain solvability results in nonresonant situations, a nonlinear Fredholm alternative (in the spirit of Landesman and Lazer) valid in both nonresonant and strongly resonant situations and establish a multiplicity result valid in nonresonant and weakly resonant situations.

In this paper, a second-order nonlinear singular boundary value problem is presented, which is equivalent to the well-known Falkner–Skan equation. And the one-dimensional third-order boundary value problem on interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document}[0,∞) is equivalently transformed into a second-order boundary value problem on finite interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\beta , 1]$$\end{document}[β,1]. The finite difference method is utilized to solve the singular boundary value problem, in which the amount of computational effort is significantly less than the other numerical methods. The numerical solutions obtained by the finite difference method are in agreement with those obtained by previous authors.

We develop a multilevel augmentation method for solving nonlinear boundary value problems. We first describe the multilevel augmentation method for solving nonlinear operator equations of the second kind which has been proposed in our recent paper Chen et al. (2011) [16], and then apply it to solving the nonlinear two-point boundary value problems of second-order differential equations. The theoretical analysis of convergence order and computational complexity are proposed. Finally numerical experiments are presented to confirm the theoretical estimates and illustrate the efficiency of the method.

The author considers a three‐point third order boundary value problem. Properties and the structure of solutions of the third order equation are discussed. Also, a connection between the number of solutions of the boundary value problem and the structure of solutions of the equation is established. First published online: 09 Jun 2011

In this paper we propose a numerical approach to solve some problems connected with the implementation of the Newton type methods for the resolution of the nonlinear system of equations related to the discretization of a nonlinear two-point BVPs for ODEs with mixed linear boundary conditions by using the finite difference method.

Are positive solutions of finite difference boundary value problems Δhu=f(u) in Ωh, u=0 on ∂Ωh as symmetric as the domain? To answer this question we first show by examples that almost arbitrary non-symmetric solutions can be constructed. This is in striking difference to the continuous case, where by the famous Gidas–Ni–Nirenberg theorem [B. Gidas, Wei-Ming Ni, L. Nirenberg, Symmetry and related problems via the maximum principle, Comm. Math. Phys. 68 (1979) 209–243] positive solutions inherit the symmetry of the underlying domain. Then we prove approximate symmetry theorems for solutions on equidistantly meshed n-dimensional cubes: explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer. The quality of the estimates depends on whether or not f(0)

The existence of solutions to a class of two-point boundary value problems in three-ion electrodiffusion is investigated via an integro-differential formulation. Boundedness by upper and lower solutions corresponding to associated boundary value problems is considered and illustrated by Painlevé II solutions of a constrained version of the original boundary value problems.

In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state- feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target $phi^*$. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.

We study the existence of solutions for the following problem: where ƒ ϵ C[0, π], g ϵ C( R ) is bounded and has limits limu → ± ∞ g(u). We also give information on the set of ƒ for those that solution exists, relating it with the corresponding linear problem.

In this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damage-dependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging.

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 Pade approximates. The validity of the approach is verified through illustrative numerical examples

Let us consider the BVP mx″ t + g 1 x ′ t = f t , t ∈ 0, T x 0 = x T , x′ 0 = x′ T , where g1 is a continuous function. The range R 1 of the operator related to this problem is very well known. In this paper we treat the perturbed problem mx″ t + g 1 x ′ t + g 0 x t = f t , t ∈ 0, T x 0 = x T , x′ 0 = x′ T , where g0 is of pendulum type, showing that, in general, the range of the perturbed operator is not contained in R 1. This points out an important qualitative difference with respect to the case where g0 is of the Landesmann–Lazer type. On the other hand we prove that if f is small then the mentioned inclusion is true in general.

In this paper, we report on a new and in some senses nontraditional way to apply Green's matrices to a numerical solution of boundary value problems for partial differential equations. We attempt to demonstrate the computational effectiveness of a semianalytical formulation for solution of external boundary value problems, modelling the contact phenomenon by a two-dimensional theory of elasticity. Linear and nonlinear statements that relate to problems with given and unknown zones of contact, respectively, have been succesfully treated by means of the proposed approach. Effective compact representations of the Green's matrices needed for this type of mathematical modelling have been obtained herein by using the technique previously developed by the authors. The paper will review a limited number of validation examples.

A new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems. An artificial parameter is introduced in the scheme. Symbolic computation is proposed for the construction of the scheme. Local truncation error of the method is discussed. Numerical examples are illustrated. Numerical results show that the method is very effective. (C) 2011 Elsevier Inc. All rights reserved.

This paper is concerned with periodic boundary value problems for a kind of first order impulsive differential equations. Some new results related to the existence of solutions are obtained by the ideas involve differential inequalities and fixed point theorems.

The Hele-Shaw flows with nonlinear kinetic undercooling regularization were discussed. The velocity vector field v ≡ (vx, vy)of a two-dimensional flow driven by a single sink at the origin with continuous intensity q0 = q0(t) was found. The nonlinear system of equations for the coefficients h̃n was reported.

Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian iterative method is three, and we design the base method in a way that we can maximize the convergence order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m - 2) for m = 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The claimed convergence order is verified by computing the computational order of convergence for a system of nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving many nonlinear initial and boundary value problems.