Order Boundary Value Problems Research Articles

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>

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>

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.

On Rayleigh-Ritz and Collocation Methods for Solving Second Order Boundary Value Problems of Ordinary Differential Equations

In this article, the Rayleigh-Ritz method is compared with the collocation method in solving second order boundary value problems of ordinary differential equations with the associated boundary conditions. The trial solution for the Rayleigh-Ritz method has to be chosen in such a way that the linearly independent functions must satisfy the boundary conditions. The collocation method, on the other hand make use of any basis function as the trial solution. The trial solution is then made to satisfy the differential equation and the boundary conditions at some interior points in the solution interval. Results obtained using the two methods show that the collocation method is simpler, easier and more accurate than the Rayleigh-Ritz method.

Numerical computational approach for 6th order boundary value problems

This study introduces numerical computational methods that employ fourth-kind Chebyshev polynomials as basis functions to solve sixth-order boundary value problems. The approach transforms the BVPs into a system of linear algebraic equations, expressed as unknown Chebyshev coefficients, which are subsequently solved through matrix inversion. Numerical experiments were conducted to validate the accuracy and efficiency of the technique, demonstrating its simplicity and superiority over existing solutions. The graphical representation of the method's solution is also presented.

Thermophoretic particle deposition on double-diffusive Ree-Eyring fluid flow across two deformable rotating disks with Hall current and Ion slip

This work emphasizes the characteristics of thermophoretic particle deposition on Ree-Eyring fluid flow with Hall current and Ion slip effects between two deformable spinning disks. The novelty of the envisioned model is strengthened by considering generalized Fourier and Fick laws with thermal source/sink. The proposed model is optimized by taking multiple slip boundary constraints at the surface. The equations that govern the flow are simplified with the use of the proper transformations. The numerical solution is obtained using the boundary value problem of fourth order (bvp4c) method. Graphical representations of the emerging parameters are presented. Numerical values of skin friction coefficient, local Nusselt, and Stanton number are highlighted in tabular form. The velocity field depicts a deteriorating behavior on incrementing the Reynold number. For augmented values of Hall and ion slip parameters fluid velocity elevates. An upsurge is noticed in the axial and radial velocities on amplifying the rotation parameter. On augmenting the thermophoretic parameter the particle deposition at the interface of the upper disk diminishes. Growing values of rotation parameter, Weissenberg, and Reynold numbers imply an additional resistance to the fluid flow that ultimately reinforced the surface drag force causing a decline in the fluid velocity. The rate of heat transmission accelerates on enhancing the Reynold number, however, heat flux exhibits a decay on augmenting the thermal slip parameter. On augmenting the thermophoretic parameter particle deposition deteriorates at the interface of the upper disk. An excellent concordance is noticed when the outcomes are contrasted with the previous literature. For the value of the rotation parameter β=-1 and β=-8 the percentage error of the present analysis with Imtiaz et al. [68] are 0.00001 and 0.00002.

A comprehensive review on fractional-order optimal control problem and its solution

Abstract This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

Cubic Trigonometric B-spline Method for Solving a Linear System of Second Order Boundary Value Problems

This paper introduces a novel trigonometric B-spline collocation method for solving a specific class of second-order boundary value problems. The study showcases the method’s practicality and effectiveness through various numerical examples. Furthermore, it evaluates the technique’s performance by calculating maximum errors for different step sizes in the spatial domain. The paper also conducts a comparative analysis with alternative methods, demonstrating the superior accuracy of the trigonometric B-spline approach.

Variational iteration algorithm for numerical solutions of sixth and seventh order boundary value problems using shifted Vieta-Lucas polynomials

This study employs Shifted Vieta-Lucas Polynomials using the variational iteration approach to numerically resolve sixth and seventh order Boundary Value Problems (BVPs), The proposed method in the study is used, with the trial functions for the approximation being the shifted Vieta Lucas polynomials generated for the given boundary value problem. The study provides an efficient and accurate solutions to higher-order BVPs using a combination of Shifted Vieta-Lucas Polynomials and the variational iteration approach and finds applications in mathematics and other fields of sciences and engineering.

Multiple solutions of a nonlinear elastic beam equation supported at two ends

In this paper, several new existence theorems on pairs of solutions are obtained for the fourth order boundary value problem ${u^{(4)}}(t) = h(t,u(t))$ subject to $u(0) = u(1) = u''(0) = u''(1) = 0$. Further, if $h(t,u) = \mu f(t,u)$, we prove that there exists $ \mu ^ * > 0$ such that there is no solution to the above boundary value problem for $\mu > {\mu ^ * }$, and that there are multiple solutions of the above boundary value problem for $0 < \mu < {\mu^ * }$.

Potentiality of the HOFiD_bvp code in solving different kind of second order boundary value problems

The aim of this paper is to present the potentiality of the code HOFiD_bvp based on high order finite difference, following the approach of the boundary value methods and the upwind methods. Strategies, as variable stepsize and variable order, that helps to handle with solutions of singular perturbation problems, so as the continuation strategy applied for solving nonlinear problems, will be discussed. Now, the method is suitable to solve second order scalar boundary value problems, not only singularly perturbed with one and two parameters, but also problems with discontinuous source terms. Concerning that point some numerical tests will be taken in consideration in order to show the potentiality of the code.

Numerical method to solve generalized nonlinear system of second order boundary value problems: Galerkin approach

In this study, we consider the system of second order nonlinear boundary value problems (BVPs). We focus on the numerical solutions of &#x0D; different types of nonlinear BVPs by Galerkin finite element method (GFEM). First of all, we originate the generalized formulation of GFEM for those type of problems. Then we determine the approximate solutions of a couple of second order nonlinear BVPs by GFEM. The approximate results are unfolded in tabuler form and portrayed graphically along with the exact solutions. Those results demonstrate the applicability, compatibility and accuracy of this scheme.

Existence for Two-Point nth Order Boundary Value Problems under Barrier Strips

Using barrier strip conditions, we study the solvability of two-point boundary value problems for the equation x(n)=f(t,x,x′,…,x(n−1)). In the case n=4, we apply the used approach to obtain results guaranteeing positive or non-negative, monotone, convex solutions to boundary value problems with various boundary conditions.