Singular Boundary Value Problems Research Articles

A Pair of Optimized Nyström Methods with Symmetric Hybrid Points for the Numerical Solution of Second-Order Singular Boundary Value Problems

This paper introduces an efficient approach for solving Lane–Emden–Fowler problems. Our method utilizes two Nyström schemes to perform the integration. To overcome the singularity at the left end of the interval, we combine an optimized scheme of Nyström type with a set of Nyström formulas that are used at the fist subinterval. The optimized technique is obtained after imposing the vanishing of some of the local truncation errors, which results in a set of symmetric hybrid points. By solving an algebraic system of equations, our proposed approach generates simultaneous approximations at all grid points, resulting in a highly effective technique that outperforms several existing numerical methods in the literature. To assess the efficiency and accuracy of our approach, we perform some numerical tests on diverse real-world problems, including singular boundary value problems (SBVPs) from chemical kinetics.

Advanced numerical scheme and its convergence analysis for a class of two-point singular boundary value problems

In the past decades, many applications related to applied physics, physiology and astrophysics have been modelled using a class of two-point singular boundary value problems (SBVPs). In this article, a novel approach based on the shooting projection method and the Legendre wavelet operational matrix formulation for approximating a class of two-point SBVPs with Dirichlet and Neumann–Robin boundary conditions is proposed. For the new approach, an initial guess is postulated in contrast to the boundary conditions in the first step. The second step deals with the usage of the Legendre wavelet operational matrix method to solve the initial value problem (IVP). Further, the resulting solution of the IVP is utilized at the second endpoint of the domain of a differential equation in a shooting projection method to improve the initial condition. These two steps are repeated until the desired accuracy of the solution is achieved. To support the mathematical formulation, a detailed convergence analysis of the new approach is conducted. The new approach is tested against some existing methods such as various types of the variational iteration method, considering several numerical examples to which it provides high-quality solutions.

Some novel analyses of the Caputo-type singular three-point fractional boundary value problems

Some novel analyses of the Caputo-type singular three-point fractional boundary value problems

Quasilinear bessel polynomial collocation method for the solutions of nonlinear singular boundary value problems

The Lane–Emden–Fowler-type equations along with boundary conditions form nonlinear singular boundary value problems. Analytical solutions to this type of problem are not easy, which draws the attention of the researchers toward their numerical solutions. Many methods have been used to provide numerical solutions to this type of problem. In this study, we have proposed the quasilinear Bessel polynomial collocation method (QBPCM) for the solution of these problems. The accuracy and efficiency of the QBPCM have been demonstrated by solving several nonlinear singular models that arise in real-life problems. It has been shown that the numerical results obtained by the proposed method have excellent agreement with the exact solutions and better accuracy than other methods.

A hybrid reptile search algorithm and Levenberg–Marquardt algorithm based Haar wavelets to solve regular and singular boundary value problems

A hybrid reptile search algorithm and Levenberg–Marquardt algorithm based Haar wavelets to solve regular and singular boundary value problems

Singular and Nonsingular Third-Order Periodic Boundary Value Problems with Indefinite Weight

Singular and Nonsingular Third-Order Periodic Boundary Value Problems with Indefinite Weight

Hybrid Cubic B-spline Method for Solving A Class of Singular Boundary Value Problems

In this paper, a class of singular two-point boundary value problems are solved using hybrid cubic B-spline method. In this algorithm, a free parameter γ plays an important role to give accurate converge results for the solution. This parameter are chosen via optimization. Numerical examples are displayed to prove that our suggested method is flexible and effective, and the numerical results are compared with other numerical methods from the literature

Weak equilibria for time‐inconsistent control: With applications to investment‐withdrawal decisions

AbstractThis paper considers time‐inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time‐inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair of control‐stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two‐dimensional example with a hyperbolic discount.

Solving singular boundary value problems using higher-order compact finite difference schemes with a novel higher-order implementation of Robin boundary conditions

This study introduces a novel and very accurate method for implementing Robin boundary conditions while solving boundary value problems with compact finite difference schemes. Most studies numerically approximate Neumann and Robin boundary conditions with a first-order approximation, however, this reduces the accuracy in general. Our implementation produces highly accurate results in solving nonlinear singular boundary value problems. Other reported results achieved utilizing various techniques are used for comparison. Tables and graphs showing the results are given.

Accurate Laguerre collocation solutions to a class of Emden–Fowler type BVP

We solve numerically the nonlinear and double singular boundary value problem formed by the well known Emden–Fowler equation , s > 1 along with the boundary conditions and In order to capture the exponential decrease of its solution we use the Laguerre–Gauss–Radau collocation method and infer its convergence. We show that the value of u ʹ at origin, which plays a fundamental role in these problems, definitely satisfies some rigorous accepted bounds. A particular attention is paid to the Thomas–Fermi case, i.e. We treat the problems as boundary values ones without any involvement of ones with initial values. The method is robust with respect to scaling and order of approximation.

Existence theory for multiple solutions to second-order singular Dirichlet boundary value problem modeling the Antarctic Circumpolar Current

In the article, we present multiple solutions for a second-order singular Dirichlet boundary value problem that arises when modeling the ocean flow of the Antarctic Circumpolar Current. The main tools of the proof are the Leray–Schauder nonlinear alternative principle and a well-known fixed point theorem in cones.

Existence of positive solutions for singular p-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term

In this paper, based on the properties of Green function and the eigenvalue of a corresponding linear operator, the existence of positive solutions is investigated by spectral analysis for a infinite-points singular p-Laplacian Hadamard fractional differential equation boundary value problem, and an example is given to demonstrate the validity of our main results.

A high-order numerical method for solving nonlinear derivative-dependent singular boundary value problems using trigonometric B-spline basis function

A high-order numerical method for solving nonlinear derivative-dependent singular boundary value problems using trigonometric B-spline basis function

Existence and Uniqueness of Positive Solutions to a Class of Singular Integral Boundary Value Problems of Fractional Order

Existence and Uniqueness of Positive Solutions to a Class of Singular Integral Boundary Value Problems of Fractional Order

A fixed point iterative scheme based on Green's function for numerical solutions of singular BVPs

<abstract><p>We suggest a novel iterative scheme for solutions of singular boundary value problems (SBVPs) that is obtained by embedding Green's function into the Picard-Mann Hybrid (PMH) iterative scheme. This new scheme we call PMH-Green's iterative scheme and prove its convergence towards a sought solution of certain SBVPs. We impose possible mild conditions on the operator or on the parameters involved in our scheme to obtain our main outcome. After this, we prove that this new iterative scheme is weak $ w^{2} $-stable. Eventually, using two different numerical examples of SBVPs, we show that our new approach suggests highly accurate numerical solutions as compared the corresponding Picard-Green's and Mann-Green's iterative schemes.</p></abstract>

Existence of solutions for a delay singular high order fractional boundary value problem with sign-changing nonlinearity

This paper consider the existence of at least one positive solution of a Riemann-Liouville fractional delay singular boundary value problem with sign-changing nonlinerty. To establish sufficient conditions we use the Guo-Krasnosel?skii fixed point theorem.

Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis

Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods.

Existence of positive solutions for singular fractional boundary value problems with p-Laplacian

In this paper, we obtain the existence of positive solutions for the singular fractional boundary value problem with p-Laplacian. The existence of positive solutions is established using the Avery-Peterson fixed point theorem. In addition, we include an example for the demonstration of our main result.

A sinc-collocation approximation solution for strongly nonlinear class of weakly singular two-point boundary value problems

In this study, an efficient collocation method based on Sinc function coupled with double exponential transformation is developed. This approach is used for solving a class of strongly nonlinear regular or weekly singular two-point BVPs with homogeneous or non homogeneous boundary conditions. The properties of the Sinc-collocation scheme were used to reduce the computations of the problem to the nonlinear system of equations. To use the Newton method in solving the nonlinear system, its vectormatrix form was obtained. The convergence analysis of the method is discussed. The analysis show that the method is convergent exponential. In order to investigate the capability and accuracy of the method, it is applied to solve several existing problems chosen from the open literature. The numerical results compared with other existing methods. The obtained results indicate high capacity and rapid convergence of the proposed method.

The bounds of eigenvalue for complex singular boundary value problems

The bounds of eigenvalue for complex singular boundary value problems