Nonlinear Singular Boundary Value Problems Research Articles

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.

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

Existence results for a class of four point nonlinear singular BVP arising in thermal explosion in a spherical vessel

In this article, the following class of four-point singular non-linear boundary value problem (NLBVP) is considered which arises in thermal explosion in a spherical vessel −(s2y′(s))′=s2f(s,y,s2y′),s∈(0,1),y′(0)=0,y(1)=δ1y(η1)+δ2y(η2), where Ω=(0,1)×R2, f:Ω→R is continuous on Ω as well as satisfy Lipschitz condition with respect to y and y′ (one sided), δ1, δ2>0 are constants, and 0<η1≤η2<1. We provide an estimation of the region of existence of a solution of above singular NLBVP. We extend the theory of monotone iterative technique (MIT) which provides computable monotone sequences that converge to the solutions of the nonlinear four point BVPs.

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.

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.

A reliable algorithm for a class of singular nonlinear two-point boundary value problems arising in physiology

In this paper, we present a reliable numerical algorithm to determine approximate solutions of the two-point boundary value problems having Robin boundary conditions that naturally occur in the investigation of distinct tumor growth issues, the dispersal of heat sources in the person head and steady state oxygen diffusion in spherical cell possessing Michaelis–Menten uptake kinetics. This approach is based on a modified concept of Adomian polynomials (AP), and the two-step Adomian decomposition method (TSADM) merged with Padé approximants. Furthermore a Maple package ADMP is applied to solve various problems, which is very easy to use and efficient and needed to input the system of equations with initial or boundary conditions and diverse essential parameters to deliver the analytic approximate solutions within a few seconds. The suggested scheme does not require linearization, perturbations, guessing the initial terms, a set of basis function or other limiting presumptions, which yields the solutions in closed form. Many examples are examined to make clear the scope and validity of the package ADMP.

Triple-Positive Solutions for a Nonlinear Singular Fractional q-Difference Equation at Resonance

Fractional q-calculus plays an extremely important role in mathematics and physics. In this paper, we aim to investigate the existence of triple-positive solutions for nonlinear singular fractional q-difference equation boundary value problems at resonance by means of the fixed-point index theorem and the q-Laplace transform, where the nonlinearity f(t,u,v) permits singularities at t=0,1 and u=v=0. The obtained theorem is well illustrated with the aid of an example.

A novel approach based on mixed exponential compact finite difference and OHA methods for solving a class of nonlinear singular boundary value problems

This work aims to find the numerical solution to a class of nonlinear singular boundary value problems (SBVPs). The considered problem has a singularity at x = 0. We introduce a computational technique comprising an optimal homotopy analysis (OHA) approach and exponential compact finite difference method (ECFDM) to solve this SBVPs. In this technique, the domain of the problem is divided into two subintervals as (the point is chosen sufficiently close to the singularity). In interval , we employ the OHA scheme to overcome the singularity. In interval , an ECFDM is designed to solve the resultant boundary value problem (BVP). Convergence analysis of the ECFDM is discussed. Furthermore, numerical experiments are performed to confirm the theoretical claims. The proposed ECFDM is shown to be fourth-order convergence.

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.

On the Computational Study of a Fully Wetted Longitudinal Porous Heat Exchanger Using a Machine Learning Approach.

The present study concerns the modeling of the thermal behavior of a porous longitudinal fin under fully wetted conditions with linear, quadratic, and exponential thermal conductivities surrounded by environments that are convective, conductive, and radiative. Porous fins are widely used in various engineering and everyday life applications. The Darcy model was used to formulate the governing non-linear singular differential equation for the heat transfer phenomenon in the fin. The universal approximation power of multilayer perceptron artificial neural networks (ANN) was applied to establish a model of approximate solutions for the singular non-linear boundary value problem. The optimization strategy of a sports-inspired meta-heuristic paradigm, the Tiki-Taka algorithm (TTA) with sequential quadratic programming (SQP), was utilized to determine the thermal performance and the effective use of fins for diverse values of physical parameters, such as parameter for the moist porous medium, dimensionless ambient temperature, radiation coefficient, power index, in-homogeneity index, convection coefficient, and dimensionless temperature. The results of the designed ANN-TTA-SQP algorithm were validated by comparison with state-of-the-art techniques, including the whale optimization algorithm (WOA), cuckoo search algorithm (CSA), grey wolf optimization (GWO) algorithm, particle swarm optimization (PSO) algorithm, and machine learning algorithms. The percentage of absolute errors and the mean square error in the solutions of the proposed technique were found to lie between to and to , respectively. A comprehensive study of graphs, statistics of the solutions, and errors demonstrated that the proposed scheme’s results were accurate, stable, and reliable. It was concluded that the pace at which heat is transferred from the surface of the fin to the surrounding environment increases in proportion to the degree to which the wet porosity parameter is increased. At the same time, inverse behavior was observed for increase in the power index. The results obtained may support the structural design of thermally effective cooling methods for various electronic consumer devices.

A novel approach for solving nonlinear singular boundary value problems arising in various physical models

A novel approach for solving nonlinear singular boundary value problems arising in various physical models

A highly accurate and computationally efficient technique for solving the electrohydrodynamic flow in a circular cylindrical conduit

The present work proposes a new, highly accurate and efficient computational technique for numerical solution of a strongly nonlinear singular two-point boundary value problem which governs electrohydrodynamic flow (EHF) of a fluid in an ion drag configuration in a circular cylindrical conduit. The nonlinearity confronted in the underlying problem is in the form of rational function and the governing equation has a singularity at the point z=0. The proposed method tackles the singularity behavior at the origin. Analysis of the proposed method is carried out. Numerical experiments were performed to demonstrate the applicability and robustness of the new method. The velocity field of EHF of a fluid in radial direction is computed. In order to justify the advantage of the method, the computed results are compared with the results obtained from some recently developed computational techniques in the literature. The effects of two relevant parameters, namely the strength of nonlinearity (σ) and the Hartmann electric number (H) on the velocity profile are investigated.

On the choice of denominator functions and convergence of NSFD schemes for a class of nonlinear SBVPs

We propose two novel non-standard finite difference (NSFD) schemes for a class of non-linear singular boundary value problems (SBVPs). One of the basic idea of NSFD schemes is that the step size Δt is replaced by a non-linear function, e.g., sinΔt, tanΔt etc. In the present work, we propose to include one more parameter in the non-linear denominator functions (DFs), e.g., in place of tanΔt, we propose to use ktan(Δt/k), where k∈[c,∞) with c∈R+ which further accelerates the accuracy and reduces the CPU time. We have applied this idea to solve generalized class of non-linear doubly singular BVPs. We observe that there exist some optimal values of k that give much better results. We determine the value of k by experimenting with various DFs on the proposed NSFD schemes. Further it will be interesting to see if we can find out optimal value of k analytically and whether this optimal value is unique. Convergence of the proposed NSFD schemes has also been proved. The results have been verified by several test examples. The results have been compared with other existing methods and matlab bvp4c solver, which shows the applicability and effectiveness of the proposed NSFD schemes.

A fourth-order numerical method for solving a class of derivative-dependent nonlinear singular boundary value problems

In this paper, a high order numerical scheme based on quartic B-spline functions is proposed to solve derivative-dependent nonlinear singular boundary value problems. Convergence of the method is analyzed. Five test problems are considered to illustrate the accuracy and efficiency of the method. The results obtained by present method are compared with those obtained by the uniform mesh cubic B-spline collocation (UCS) method [1] and non-uniform mesh cubic B-spline collocation (NCS) method [1]. The CPU time of proposed method is compared with that of the NCS method.

A novel computing stochastic algorithm to solve the nonlinear singular periodic boundary value problems

In this work, a class of singular periodic nonlinear differential systems (SP-NDS) in nuclear physics is numerically treated by using a novel computing approach based on the Gudermannian neural networks (GNNs) optimized by the mutual strength of global and local search abilities of genetic algorithms (GA) and sequential quadratic programming (SQP), i.e. GNNs-GA-SQP. The stimulation of offering this numerical computing work comes from the aim of introducing a consistent framework that has an effective structure of GNNs optimized with the backgrounds of soft computing to tackle such thought-provoking systems. Two different problems based on the SPNDS in nuclear physics will be examined to check the proficiency, robustness and constancy of the GNNs-GA-SQP. The outcomes obtained through GNNs-GA-SQP are compared with the true results to find the worth of designed procedures based on the multiple trials.

A Novel Neuroevolutionary Paradigm for Solving Strongly Nonlinear Singular Boundary Value Problems in Physiology

In the present research work, we designed a hybrid stochastic numerical solver to investigate nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions arising in various physical models. In this method, we hybridized Harris Hawks Optimizer with Interior Point Algorithm named HHO-IPA. We construct artificial neural networks (ANNs) model for the problem, and this model is tuned with the proposed scheme. This scheme overcomes the singular behavior of problems. The accuracy and applicability of the method are illustrated by finding absolute errors in the solution. The outcomes are compared with the results present in the literature to demonstrate the effectiveness and robustness of the scheme by considering four different nonlinear singular boundary value problems. Further, the convergence of the scheme is proved by performing computational complexity analysis. Moreover, the graphical overview of statistical analysis is added to our investigation to elaborate further on the scheme’s stability, accuracy, and consistency.

Lucas polynomials based spectral methods for solving the fractional order electrohydrodynamics flow model

The aim of this paper is to study the fractional order nonlinear singular boundary value problem arising in electrohydrodynamics flow, which describes the velocity of the ionized fluid in a circular cylindrical conduit. For this purpose, Lucas polynomials are employed via spectral collocation and Galerkin methods for reducing the nonlinear singular differential equation of fractional order to an algebraic nonlinear system, which is solved by Newton iterative method. The problem includes two important parameters, including the nonlinearity and Hartman numbers. The effect of these parameters and the influence of fractional order derivatives on the velocity of the fluid is discussed. Based on the reported results in tables and figures, for large values of Hartman number and the fractional derivatives, as well as weak nonlinearity parameter, the velocity profile was increased. The purposed schemes are simple and attractive, and their accuracy and efficiency, were confirmed by studying some cases of main problem and a comparison of the numerical results with the results obtained by some other methods.

A quartic trigonometric B-spline collocation method for a general class of nonlinear singular boundary value problems

A quartic trigonometric B-spline collocation method for a general class of nonlinear singular boundary value problems

An efficient numerical approach for solving a general class of nonlinear singular boundary value problems

An efficient numerical approach for solving a general class of nonlinear singular boundary value problems

A note on variation iteration method with an application on Lane–Emden equations

PurposeIn this article, the authors consider the following nonlinear singular boundary value problem (SBVP) known as Lane–Emden equations, −u″(t)-(α/t) u′(t) = g(t, u), 0 < t < 1 where α ≥ 1 subject to two-point and three-point boundary conditions. The authors propose to develop a novel method to solve the class of Lane–Emden equations.Design/methodology/approachThe authors improve the modified variation iteration method (VIM) proposed in [JAAC, 9(4) 1242–1260 (2019)], which greatly accelerates the convergence and reduces the computational task.FindingsThe findings revealed that either exact or highly accurate approximate solutions of Lane–Emden equations can be computed with the proposed method.Originality/valueNovel modification is made in the VIM that provides either exact or highly accurate approximate solutions of Lane-Emden equations, which does not exist in the literature.