Singular Boundary Value Problems Research Articles

An efficient fourth-order convergent scheme based on half-step spline function for two-point mixed boundary value problems

The present work aims to introduce a novel numerical method for solving second-order two-point mixed boundary value problems. Mixed boundary value problems occur in various scientific fields, including quantum mechanics, fluid dynamics, and chemical reactor theory. The proposed method provides a highly accurate, fourth-order convergent numerical solution, achieved by implementing a compact finite difference method based on non-polynomial spline in tension approximations. Notably, the discretisation involves the use of half-step grid points, eliminating the need to modify the method at singularities while dealing with singular boundary value problems. The article also includes a comprehensive convergence analysis that demonstrates the theoretical order of convergence of the method. Additionally, the results obtained from solving six diverse mixed boundary value problems, including Burger's equation and Bratu-type equation, are compared to showcase the superiority and effectiveness of the proposed method.

ANN based optimization of nano-beam oscillations with intermolecular forces and geometric nonlinearity

In this study, we investigate the effect of Van der Waals and Casimir forces on the mathematical model of nano-electromechanical systems (NEMS) such as nano-beam actuators that contain cantilever and double cantilever beams. The singular nonlinear boundary value problem governing the beam-type actuators, including geometric nonlinearity is solved by using an intelligent strength of feedforward artificial neural networks (ANNs) and hybridization of optimization algorithms such as arithmetic optimization algorithm (AOA) and active set algorithm (ASA). The proposed ANN-AOA-AS algorithm is employed to quantify the effect of changes in applied voltage, dispersion forces, geometric nonlinearity parameters, and initial axial strain on the deflection of the beam. Furthermore, to validate the results obtained by the proposed algorithm, statistical analyses are conducted to compare the approximate solutions with state-of-the-art methodologies available in the latest literature. In addition, performance indicators are defined such as mean square error (MSE), Nash–Sutcliffe efficiency (NSE), mean absolute deviations (MAD), root mean square error (RMSE), and Error in Nash–Sutcliffe efficiency (ENSE) to study the accuracy and efficiency of the solutions. The results show that these indicators’ mean percentage values lie around 10−4 to 10−6 which reflects the perfect modeling of the approximate solutions.

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.

Nonlinear Contractions Employing Digraphs and Comparison Functions with an Application to Singular Fractional Differential Equations

After the initiation of Jachymski’s contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem.

Spectral Solutions of Specific Singular Differential Equations Using A Unified Spectral Galerkin-Collocation Algorithm

This paper presents a novel numerical approach to addressing three types of high-order singular boundary value problems. We introduce and consider three modified Chebyshev polynomials (CPs) of the third kind as proposed basis functions for these problems. We develop new derivative operational matrices for the three modified CPs of the third kind by deriving formulas for their first derivatives. Our approach follows a unified method for numerically handling singular differential equations (DEs). To transform these equations into algebraic systems suitable for numerical treatment, we employ the collocation method in combination with the introduced operational matrices of derivatives of the modified CPs of the third kind. We address the convergence examination for the three expansions in a unified manner. We present numerous numerical examples to demonstrate the accuracy and efficiency of our unified numerical approach.

Analytical method for systems of nonlinear singular boundary value problems

The standard Lane–Emden equations model several physical phenomena such as isotropic continuous media, thermal behaviour of a spherical cloud of gas, and isothermal gas spheres. Systems of Lane–Emden-type equations appear in the modelling of the concentration of carbon substrate and oxygen, catalytic diffusion reactions, the steady state concentration of carbon dioxide, dusty fluid models, and pattern formation. In solving singular boundary value problems, one faces challenges resulting from the divergence of the associated variable coefficients at the singular points. This research article explores the power series approach to analytically approximate solutions to a class of systems of strongly nonlinear singular boundary value problems of Lane–Emden-type. The nonlinear terms in the proposed problems are transformed into power series using the generalised Cauchy product before establishing explicit recursion formulae for the expansion coefficients of the system of series solutions. The initial conditions required in the proposed boundary value problems are assumed and determined from a set of nonlinear algebraic equations resulting from the given right boundary conditions. Three special systems of nonlinear singular boundary value problems of Lane–Emden type are presented to demonstrate the proposed method’s reliability, effectiveness, and accuracy. The obtained approximate solutions are compared with the exact solutions (where they are available), or with other existing results (where the exact solution is not readily available). The series solution of the first example has a slow convergent rate, while the results of the other two examples are in excellent agreement with the exact solutions and other published results.

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.

Bernstein collocation technique for a class of Sturm-Liouville problems

Sturm-Liouville problems have yielded the biggest achievement in the spectral theory of ordinary differential operators. Sturm-Liouville boundary value issues appear in many key applications in natural sciences. All the eigenvalues for the standard Sturm-Liouville problem are guaranteed to be real and simple, and the related eigenfunctions form a basis in a suitable Hilbert space. This article uses the weighted residual collocation technique to numerically compute the eigenpairs of both regular and singular Strum Liouville problems. Bernstein polynomials over [0,1] has been used to develop a weighted residual collocation approach to achieve an improved accuracy. The properties of Bernstein polynomials and the differentiation formula based on the Bernstein operational matrix are used to simplify the given singular boundary value problems into a matrix-based linear algebraic system. Keeping this fact in mind such a polynomial with space defined collocation scheme has been studied for Strum Liouville problems. The main reasons to use the collocation technique are its affordability, ease of use, well-conditioned matrices, and flexibility. The weighted residual collocation method is found to be more appealing because Bernstein polynomials vanish at the two interval ends, providing better versatility. A multitude of test problems are offered along with computation errors to demonstrate how the suggested method behaves. The numerical algorithm and its applicability to particular situations are described in detail, along with the convergence behavior and precision of the current technique.

A high-order B-spline collocation method for solving a class of nonlinear singular boundary value problems

A high-order B-spline collocation method for solving a class of nonlinear singular boundary value problems

VISCOSITY SOLUTIONS TO THE INFINITY LAPLACIAN EQUATION WITH SINGULAR NONLINEAR TERMS

Abstract In this paper, we study the singular boundary value problem $$ \begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$ where $\lambda>0$ is a parameter, $h>1$ and $\Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator related to the infinity Laplacian. The nonlinear term $f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is a continuous function and may exhibit singularity at $t\rightarrow 0^{+}$ . We establish the comparison principle by the double variables method for the general equation $\Delta _\infty ^h u=F(x,u,Du)$ under some conditions on the term $F(x,t,p)$ . Then, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron’s method and the comparison principle. Finally, we obtain the existence result in the entire Euclidean space by the approximation procedure. In this procedure, we also establish the local Lipschitz continuity of the viscosity solution.

Collocation Technique Based on Chebyshev Polynomials to Solve Emden–Fowler-Type Singular Boundary Value Problems with Derivative Dependence

In this work, an innovative technique is presented to solve Emden–Fowler-type singular boundary value problems (SBVPs) with derivative dependence. These types of problems have significant applications in applied mathematics and astrophysics. Initially, the differential equation is transformed into a Fredholm integral equation, which is then converted into a system of nonlinear equations using the collocation technique based on Chebyshev polynomials. Subsequently, an iterative numerical approach, such as Newton’s method, is employed on the system of nonlinear equations to obtain an approximate solution. Error analysis is included to assess the accuracy of the obtained solutions and provide insights into the reliability of the numerical results. Furthermore, we graphically compare the residual errors of the current method to the previously established method for various examples.

Smooth Lyapunov Manifolds for Autonomous Systems of Nonlinear Ordinary Differential Equations and Their Application to Solving Singular Boundary Value Problems

Smooth Lyapunov Manifolds for Autonomous Systems of Nonlinear Ordinary Differential Equations and Their Application to Solving Singular Boundary Value Problems

Multiple Solutions for a Class of Singular Boundary Value Problems of Hadamard Fractional Differential Systems with <i>p</i>-Laplacian Operator

Multiple Solutions for a Class of Singular Boundary Value Problems of Hadamard Fractional Differential Systems with <i>p</i>-Laplacian Operator

A matrix technique‐based numerical treatment of a nonlocal singular boundary value problems

The mathematical modeling of the decisive event of astrophysics, physiology, and many other areas of science and technology witness the involvement of singular boundary value problems. The nonlocal boundary conditions are more informative than local boundary conditions (initial conditions and two‐point boundary conditions) to evaluate some mathematical models. This article presents a collocation approach‐based matrix technique to approximate the solution of the fusion of a class of singular differential equations subject to nonlocal three‐point boundary conditions. The proposed strategy utilizes the truncation of the series expansion of a function belonging to in terms of Bernoulli polynomials. It transforms the singular boundary value problems into a set of nonlinear algebraic equations, which can be dealt with by any mathematical software. The Lipschitz condition on an equivalent completely continuous nonlinear operator has been used to prove the convergence analysis of the scheme. Some extremely nonlinear test examples are solved and provided in contrast with the exact solution. These numerical results are also examined against some existing numerical techniques to verify the applicability and significance of the proposed methodology. There are a few numerical examples that are application based but do not have exact solutions. In such cases, residual error norm is employed to measure the accuracy of the numerical strategies. The computational data demonstrate the superiority and validity of the proposed technique over existing numerical approaches.

An accurate numerical method of solving singular boundary value problems for the stationary flow of granular materials and its application

An accurate numerical method of solving singular boundary value problems for the stationary flow of granular materials and its application

On the numerical integration of singular initial and boundary value problems for generalised Lane–Emden and Thomas–Fermi equations

We propose a geometric approach for the numerical integration of singular initial and boundary value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane–Emden equations and the Thomas–Fermi equation.

A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better.

Order-preserving fuzzy transform for singular boundary value problems of polytropic gas flow and sewage diffusion

We describe a fuzzy transform method to analyze two-point singular and non-singular boundary value problems with high-order solution accuracies. This framework implements a fuzzy transform that approximates solutions with fourth-order accuracy at the interior mesh points. The fuzzy components and the triangular base function are locally arranged with a three-point linear combination of solution values. This yields a tri-diagonal Jacobian matrix that can be easily computed in a space-time efficient manner. Since a linear system relates solution values and fuzzy components, it is easy to obtain solution approximations via fuzzy components by a tri-diagonal matrix inversion. In addition to the numerical solution, it is easy to determine an approximate analytic solution using a cubic spline interpolating polynomial from the data available with fuzzy components. The error estimates for approximate analytical solutions and numerical solutions are obtained by integrated absolute error and maximum absolute errors. The new mechanism is analyzed for convergence using matrix theory. Several linear and nonlinear equations of practical importance related to sewage diffusion and polytropic gas flow models are simulated to corroborate the new scheme's utility and fourth-order convergence. Mathematics subject classification03B80, 65L10, 65L12

Numerical solution of third order singular boundary value problems with nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation

In this paper, we use the nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation methods for the numerical solution of singular third-order boundary value problems. The novelty of the approach is based on using the new non-classical weight function for sinc method instead of the classic ones. The convergence and error estimation of our methods is discussed. Several examples are solved and numerical results are compared with the existed methods, the obtained results demonstrate the validity of the obtained theoretical results and the efficiency of the our methods.

A new algorithm of boundary value problems based on improved wavelet basis and the reproducing kernel theory

Relying upon Legendre polynomials, an improved multiwavelet orthonormal basis is constructed in the reproducing kernel space. We study boundary value problems (BVPs) based on the improved wavelet orthonormal basis and construct the ‐best approximate solution. Convergence and the stability are discussed; this method has higher order convergence. The main advantage of the wavelet based our method is its efficiency and simple applicability for a variety of boundary conditions. Some numerical tests even including singular BVPs reveal the stability and efficiency of the new algorithm.