Second Order Boundary Value Problems

In the literature so far, for Caputo fractional boundary value problems of order 2q when 1<2q<2, the problems use the same boundary conditions of the integer-order differential equation of order ‘2’. In addition, they only use the left Caputo derivative in computing the solution of the Caputo boundary value problem of order 2q. Further, even the initial conditions for a Caputo fractional differential equation of order nq use the corresponding integer-order initial conditions of order ‘n’. In this work, we establish that it is more appropriate to use the Caputo fractional initial conditions and Caputo fractional boundary conditions for sequential initial value problems and sequential boundary value problems, respectively. It is to be noted that the solution of a Caputo fractional initial value problem or Caputo fractional boundary value problem of order ‘nq’ will only be a Cnq solution and not a Cn solution on its interval. In this work, we present a methodology to compute the solutions of linear sequential Caputo fractional differential equations using initial and boundary conditions of fractional order kq, k=0,1,…(n−1) when the order of the fractional derivative involved in the differential equation is nq. The Caputo left derivative can be computed only when the function can be expressed as f(x−a). Then the Caputo right derivative of the same function will be computed for the function f(b−x). Further, we establish that the relation between the Caputo left derivative and the Caputo right derivative is very essential for the study of Caputo fractional boundary value problems. We present a few numerical examples to justify that the Caputo left derivative and the Caputo right derivative are equal at any point on the Caputo function’s interval. The solution of the linear sequential Caputo fractional initial value problems and linear sequential Caputo fractional boundary value problems with fractional initial conditions and fractional boundary conditions reduces to the corresponding integer initial and boundary value problems, respectively, when q=1. Thus, we can use the value of q as a parameter to enhance the mathematical model with realistic data.

Background/Objectives: Numerically solving second order Boundary Value Problems (BVPs) directly is an area that has drawn much attention in recent literature and hence the general aim of this article. Methods/Statistical Analysis: Block methods are developed using Taylor series approach and are applied to directly approximate the considered second order Boundary Value Problems. Certain specifications of the block method developed involves it being of step number k = 4 and order p = 5. This block method consists of mathematical expressions that will simultaneously provide results at the grid points and derivatives of the grid points for the k-step method. Findings: After the development of the new block method in this article, certain Boundary Value Problems are considered and the new block method is adopted to solve it numerically and comparison was made with their exact solution. The results show the new block method show superiority in comparison to other numerical methods of same order p = 5 adopted by other authors in previous literature. Hence, this new block method can be considered a more suitable numerical approach to approximate the numerical solutions of second order BVPs. Applications/Improvements: This block method is suitable for application in solving numerically second order BVPs with closer accuracy to the exact or analytic solutions of the differential equations under consideration.

In this work, a technique based on combination between collocation and spline method along with the shooting method is proposed for the solution of boundary value problem of order 2. Numerical results show that the method is simple and effective.

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

Numerous papers deal with asymptotic methods for boundary value problems in domains depending on a small parameter in a complicated way (perforated domains, partially perforated domains, framework structures, thin domains); e.g., see [1–9]. Boundary value problems in thick singularly degenerating junctions (the number of components of such junctions grows infinitely as the perturbation parameter tends to zero) have specific difficulties and require a separate consideration. It was shown in [10] that boundary value problems in thick junctions lose their coercivity under the passage to the limit, which substantially complicates asymptotic studies. The papers [11, 12] were the first in this direction. In [13–19], a classification of thick junctions was given and rigorous mathematical methods were developed for analyzing the main boundary value problems of mathematical physics in thick singularly degenerating junctions of various types. The study of boundary value problems in thick junctions is focused on the asymptotic behavior of their solutions as e → 0, i.e., as the number of thin attached domains grows infinitely and their thickness tends to zero. In the present paper, we consider a thick junction whose thin attached cylinders are divided into two levels depending on the boundary conditions posed on the lateral surface of these cylinders (inhomogeneous Neumann boundary conditions and homogeneous Dirichlet conditions). In addition, thin cylinders of each level e-periodically alternate along the junction zone. Such thick junctions will be called thick two-level junctions. A problem on a thick plane two-level junction was considered for the first time in [20], where the asymptotic behavior of eigenvalues and eigenfunctions of a spectral boundary value problem was analyzed. Other boundary value problems in plane two-level junctions were considered in [21–23]. Asymptotic analysis of boundary value problems with a periodic change of boundary conditions (Neumann and Dirichlet conditions) on the boundary of smooth unperturbed domains was carried out in [24–26], where it was shown that the first term of the asymptotics is mainly the solution of the corresponding boundary value problem with the Dirichlet conditions. The qualitatively new result of the present paper implies that the first term of the asymptotics is a vector function whose components are solutions of two independent boundary value problems (one in the junction body and another in a domain filled with thin cylinders in the limit) with homogeneous Dirichlet conditions in the junction zone. Note also that the second problem is a problem for a second-order ordinary differential equation with a new right-hand side that “remembers” the inhomogeneity in the Neumann boundary conditions of the original problem and the specific “packing density” of thin cylinders.

In this paper, we consider nonlinear second order m-point impulsive time scale boundary value problems on infinite intervals. By using Leray-Schauder fixed point theorem, Avery-Henderson fixed point theorem and the five functional fixed point theorem, respectively, we establish the criteria for the existence of at least one, two and three positive solutions to the nonlinear impulsive time scale boundary value problems on infinite intervals.

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincare inequality of (Ammann-Grosse-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, well-known theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition.

New version of Optimal Homotopy Asymptotic Method for the solution of nonlinear boundary value problems in finite and infinite intervals

The scattering of surface water waves by bottom undulation in the presence of a permeable vertical barrier is investigated for its solution. A mixed boundary value problem (BVP) arises here in a natural way while examining this physical problem. Regular perturbation analysis is employed to determine the solution of the BVP. By utilizing this analysis the given BVP reduces to two different BVPs up to first order. The solution of the zeroth order BVP is obtained with the aid of eigenfunction expansion method in conjunction with least-squares approximation. The first order BVP is solved with the help of the Green's integral theorem and the physical quantities, namely the reflection and transmission coefficients, are obtained in the form of integrals which involve the bottom undulation and the solution of the zeroth order BVP. A particular form of the bottom undulation which closely resembles to some obstacles made by nature due to sedimentation and ripple growth of sand, is considered to evaluate these integrals. The variation of these coefficients is examined for different values of the porous effect parameter, barrier length, number of ripples and ripple amplitude.

We present two collocation algorithms based on interpolating and approximating subdivision schemes for the solution of fourth order boundary value problems arising in the mathematical modeling of viscoelastic, and inelastic flows, deformation of beams, arches, and load bearing members like street lights and robotic arms in multipurpose engineering systems. Numerical examples are given to illustrate the algorithms. We conclude that approximating schemes based collocation algorithms give better solution than interpolating schemes based collocation algorithms. Main purpose of this paper is to explore and seek the applications of interpolating and approximating subdivision schemes in the field of boundary value problems along with intrinsic comparison of the results obtained by algorithms based on these schemes. A comparison with other approaches of this type of boundary value problems in order to see the advantages of the proposed methods is also given.

In this paper, we investigate the existence of at least one, two and three positive solutions to the nonlinear second order m-point impulsive time scale boundary value problems on infinite intervals by using the Krasnosel?skii fixed point theorem, Avery-Henderson fixed point theorem and the five functionals fixed point theorem, respectively.

We study the existence of positive solutions for some nonlinear second order boundary value problems with nonlocal boundary conditions. The key boundary condition considered is of the form u(1) = α[u 0 ], where α is a linear functional on C[0, 1], that is, is given by a Riemann–Stieltjes integral α[v] = R 1 0 v(s)dA(s) where A is a function of bounded variation. It is important in our case that dA is not a positive measure but can be sign changing.

On the existence of positive solutions for the second-order boundary value problem

Symmetric positive solutions of boundary value problems with integral boundary conditions

Graded mesh modified backward finite difference method for two parameters singularly perturbed second-order boundary value problems