Order Boundary Value Problems Research Articles

Spectral Tau Method with Legendre Polynomials to Approximate Second Order Boundary Value Problem

Spectral Tau Method with Legendre Polynomials to Approximate Second Order Boundary Value Problem

A modified basis of cubic B-spline with free parameter for linear second order boundary value problems: Application to engineering problems

The traditional cubic B-spline method offers limited local control over the curve solution. Adjusting the position of a control point affects the entire curve, making it challenging to make localized changes, e.g., smoothness. Moreover, the basis functions vanish on one side by the cubic B-spline method near the end conditions where the initial and boundary conditions are applied. To address these limitations, this research proposes a new basis by including a free parameter γ with the purpose of modifying the weights of nearby control points. This free parameter γ can influence the curve’s behavior in specific regions as well as the entire curve. This modification of the cubic B-spline method was used to approximate the second-order derivative at each collocation point. The convergence test showed that the proposed method was second-order convergent. Numerical examples of ordinary differential equations were used with different step values to evaluate the accuracy of the proposed method. The findings persistently indicated that the proposed technique provided better error estimates as compared to the other methods discussed in the literatures.

Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions

Abstract In this paper, we study the existence of positive solutions for a fourth order boundary value problem coupled to functional perturbed clamped beam boundary conditions. Our main ingredient is the classical fixed point index. The problem investigated is an extension of other problems studied in previous papers by covering very general nonlocal perturbed conditions on the boundary.

A highly accurate and efficient Genocchi‐based spectral technique applied to singular fractional order boundary value problems

This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the norm is obtained via a rigorous error analysis. The main benefit of the presented QLM‐GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high‐order accuracy of the QLM‐GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well‐established available computational methods. It is apparently shown that the QLM‐GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two‐term fractional derivatives accurately and efficiently.

A sixth order kernel functions approach for nonlinear fourth order boundary value problems

A sixth order kernel functions approach for nonlinear fourth order boundary value problems

Differentiation of Solutions of Caputo Boundary Value Problems with Respect to Boundary Data

Under suitable continuity and uniqueness conditions, solutions of an α order Caputo fractional boundary value problem are differentiated with respect to boundary values and boundary points. This extends well-known results for nth order boundary value problems. The approach used applies a standard algorithm to achieve the result and makes heavy use of recent results for differentiation of solutions of Caputo fractional intial value problems with respect to initial conditions and continuous dependence for Caputo fractional boundary value problems.

Existence of solutions to multi-point boundary value problem of fractional order on the half-line

The purpose of this paper is to establish the existence of solutions to multi-point fractional boundary value problem on an infinite interval. Using the fixed point theory, sufficient conditions are obtained that guarantee the existence of at least one solution. At the end, an example is presented to illustrate the main results

Existence and uniqueness of Solution for Boundary Value Problem of Fractional Order

In this study, we investigate a class of fractional ordering and fractional derivative-based boundary value problems. and . There are four boundary value requirements in this equation. The Banach fixed point theorem (Contraction mapping theorem) and the Schauder fixed point theorem are both used to arrive at the existence and uniqueness solution. Examples based on the fractional integral method and integral operator are used to illustrate our main points.

Numerical solution of singularly perturbed singular third order boundary value problems with nonclassical sinc method

In this paper, we employ a nonclassical sinc-collocation method to compute numerical solutions for singularly perturbed singular third-order boundary value problems prevalent in various scientific and engineering domains. Utilizing the sinc approximation offers a strategic advantage in navigating singularities, thus enabling an efficient computational strategy. Our method streamlines the solution process by converting singular boundary value problems into sets of linear equations, thereby improving computational efficiency. Moreover, its straightforward implementation adds to its robustness. We explore the convergence properties and error estimation of our proposed methods in detail. Finally, we provide two illustrative examples that demonstrate the effectiveness of our approach.

A new iterative computational scheme for solving second order (1 + 1) boundary value problems with non-homogeneous Dirichlet conditions

A new iterative computational scheme for solving second order (1 + 1) boundary value problems with non-homogeneous Dirichlet conditions

Eleventh Degree Polynomial Series Solution Approach of Special Non-linear Fourth Order Boundary Value Problems

In this paper a new eleventh degree polynomial series solution approach is developed for the solution of special non-linear fourth order boundary value problems. The method consist first of obtaining a linear differentials system of twelve equations from the boundary conditions, governing equation and its three successive derivatives which were evaluated at boundary points. Secondly we assume the approximate solution in the form of a polynomial of degree eleventh with twelve unknown coefficients. To determine the unknown coefficients we incorporate the assumed solution into linear differentials systems of twelve equations which results into a linear systems of twelve equations with twelve unknown and which can be solve uniquely. It is clear from the tables and figures that the method is in good agreement with the exact and with some existing results in the literatures. Also it can be seen from example 3.3 that the exact solution is reproduced which is an added advantage of the method.

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

The existence of boundary layers in the solutions of two-parameter singularly perturbed boundary value problems makes classical numerical methods insufficient in providing accurate approximations. Consequently, the development of layer-adapted mesh methods that achieve parameter uniform convergence and are specifically designed to accurately handle these layers becomes highly significant. The aim of this paper is to construct and examine a numerical approach for obtaining approximate solutions to a specific class of two-parameter singularly perturbed second order boundary value problems whose solutions exhibit boundary layers at both ends of the domain. The problem is discretized by employing a modified backward finite difference method on a mesh that is graded. To validate the theoretical findings, well known test problems from the existing literature are utilized. Moreover, the efficiency of the proposed method is demonstrated by comparing it to other existing methods in the literature. The stability and uniform convergence with respect to the parameters of the proposed method have been verified, revealing that it attains second-order convergence in the maximum norm. The numerical outcomes illustrate that the proposed method offers remarkably accurate approximations of the solution. Theoretical findings are in agreement with the experimental results.

Analytical Solutions of Generalised Emden–Fowler Initial and Boundary Value Problems of Higher Order

Analytical Solutions of Generalised Emden–Fowler Initial and Boundary Value Problems of Higher Order

Mixed fractional order p-Laplacian boundary value problem with a two-dimensional Kernel at resonance on an unbounded domain

In this paper, by using the Ge and Ren extension of coincidence degree theory, we established the existence of a solution for a resonant mixed fractional order p-Laplacian boundary value problem (BVP) on the half-line. In the process, we solved the corresponding homogeneous fractional order BVP for conditions critical for resonance and showed that the operator A(x,λ)(t) constructed from the abstract equation Mx(t)=Nx(t) is relatively compact. The results are demonstrated with an example.

A novel iterative scheme for solving delay differential equations and third order boundary value problems via Green's functions

<abstract><p>In this paper, we constructed a novel fixed point iterative scheme called the Modified-JK iterative scheme. This iteration process is a modification of the JK iterative scheme. Our scheme converged weakly to the fixed point of a nonexpansive mapping and strongly to the fixed point of a mapping satisfying condition (E). We provided some examples to show that the new scheme converges faster than some existing iterations. Stability and data dependence results were proved for this iteration process. To substantiate our results, we applied our results to solving delay differential equations. Furthermore, the newly introduced scheme was applied in approximating the solution of a class of third order boundary value problems (BVPs) by embedding Green's functions. Moreover, some numerical examples were presented to support the application of our results to BVPs via Green's function. Our results extended and generalized other existing results in literature.</p></abstract>

ON RELATION BETWEEN NORMING CONSTANTS AND CHARACTERISTIC FUNCTION OF OPERATOR STURM–LIOUVILLE PROBLEM

Abstract. We consider the second order boundary value problem where coefficients of equation and boundary conditions are unbounded operators. Define relationship between characteristic determinant and norming constants of corresponding to problem operator.

Fractional differential equation with movable boundary conditions

In this research paper, we discuss the complex-valued solutions for the nonlinear fractional boundary value problem (FBVP) of complex order (δ = τ + ιa; 1 < τ ≤ 2, a ∈ R+) with movable boundary conditions. The fractional operators are taken in the sense of Riemann-Liouville (R-L) with complex order. By using the concept of Green’s function, the existence and uniqueness of solutions are established in this article. Also, we prove that the FBVP of complex order with movable boundary conditions is Ulam-Hyers Stable. Using illustrative examples, the results for this nonlinear FBVP have been shown.

Stability on a boundary problem with RL-Fractional derivative in the sense of Atangana-Baleanu of variable-order

<abstract><p>In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable.</p></abstract>

A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Maclaurin Series Solution for Seventh Order Boundary Value Problems

Maclaurin series expansion which is taught at undergraduate levels and which has been previously used for the solution of initial value problem is successfully used in this paper for the solutions of seventh-order boundary value problems. The governing equation defined between boundary points 0 and 1is differentiated to get high order derivatives and the assumed solution is expanded in Maclaurin series then, the boundary conditions at x=1 are utilized to determined unknown coefficients. The method is simple and the numerical results displayed on the tables were found to be in good agreement with the solution and even better than some existing results in the literature. This agreement is also confirmed from the figures were the graph of exact solution and approximate solution overlap.