Nonlinear Singular Boundary Value Problems Research Articles

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.

A boundary shape function iterative method for solving nonlinear singular boundary value problems

In this paper, a novel iterative algorithm is developed to solve second-order nonlinear singular boundary value problem, whose solution exactly satisfies the Robin boundary conditions specified on the boundaries of a unit interval. The boundary shape function is designed such that the boundary conditions can be fulfilled automatically, which renders a new algorithm with the solution playing the role of a boundary shape function. When the free function is viewed as a new variable, the original singular boundary value problem can be properly transformed to an initial value problem. For the new variable the initial values are given, whereas two unknown terminal values are determined iteratively by integrating the transformed ordinary differential equation to obtain the new terminal values until they are convergent. As a consequence, very accurate solutions for the nonlinear singular boundary value problems can be obtained through a few iterations. The present method is different from the traditional shooting method, which needs to guess initial values and solve nonlinear algebraic equations to approximate the missing initial values. As practical applications of the present method, we solve the Blasius equation for describing the boundary layer behavior of fluid flow over a flat plate, where the Crocco transformation is employed to transform the third-order differential equation to a second-order singular differential equation. We also solve a nonlinear singular differential equation of a pressurized spherical membrane with a strong singularity.

An effective approach for solving a class of nonlinear singular boundary value problems arising in different physical phenomena

In this article, an effective computational technique is developed to get an analytical approximation to the solution of a class of two-point nonlinear singular boundary value problems arising in different physical models. The proposed approach consists of two steps. Firstly, formulate an integral operator by involving Green's function, and then, the Halpern's fixed-point method is implemented to the obtained integral operator to establish the desired iterative technique. The convergence of this approach is also discussed. To show the efficiency of our approach, various numerical examples are considered. The main advantages of the proposed method over existing methods are that the proposed method does not require Adomian polynomials to handle the nonlinearity, solves the problem without using the Lagrange multipliers and constrained variations and takes both endpoints of the interval into consideration. Further, the proposed method tackles the problems without requiring linearization, discretization, and perturbation assumptions, unlike other semi-analytical methods.

Evolutionary computing for nonlinear singular boundary value problems using neural network, genetic algorithm and active-set algorithm

In this numerical study, a class of nonlinear singular boundary value problem is solved by implementation of a novel meta-heuristic computing tool based on the artificial neural networks (ANNs) modeling of system and the optimization of decision variable of ANNs through the combined strength of global search via genetic algorithms (GA) and local search ability of active-set algorithm (ASA), i.e., ANN–GA–ASA. The proposed intelligent computing solver ANN–GA–ASA exploits the input, hidden, and output layers’ structure of ANNs. This is to represent the differential model in the nonlinear singular second-order periodic boundary value problems, which are connected to form an error-based objective function (OF) and optimize the OF by the integrated heuristics of GA–ASA. The purpose to present this research is to associate the operational legacy of neural networks and to challenge such kinds of inspiring models. Two different examples of the singular periodic model have been investigated to observe the robustness, proficiency and stability of the ANN–GA–ASA. The proposed outcomes of ANN–GA–ASA are compared with reference to true results so as to establish the value of the designed scheme. Exhaustive comparison has been made and presented between the Log-sigmoidal ANNs results and the radial basis ANNs outcomes. The reliability of the results obtained is endorsed by using both types of networks as well as the value of designed schemes.

A simple, efficient and accurate new Lie--group shooting method for solving nonlinear boundary value problems

The present paper provides a new method for numerical solution of nonlinear boundary value problems. This method is a combination of group preserving scheme (GPS) and a shooting--like technique which takes advantage of two powerful methods for solving nonlinear boundary value problems. This method is very effective to search unknown initial conditions. To demonstrate the computational efficiency, the mentioned method is implemented for some nonlinear exactly solvable differential equations including strongly nonlinear Bratu equation, nonlinear reaction--diffusion equation and one singular nonlinear boundary value problem. It is also applied successfully on two nonlinear three--point boundary value problems and a third--order nonlinear boundary value problem which the exact solutions of this problems are unknown. The examples show the power of method to search for unique solution or multiple solutions of nonlinear boundary value problems with high computational speed and high accuracy. In the test problem 5 a new branch of solutions is found which shows the power of the method to search for multiple solutions and indicates that the method is successful in cases where purely analytic methods are not.

The series expansion and Chebyshev collocation method for nonlinear singular two-point boundary value problems

The solution of singular two-point boundary value problem is usually not sufficiently smooth at one or two endpoints of the interval, which leads to a great difficulty when the problem is solved numerically. In this paper, an algorithm is designed to recognize the singular behavior of the solution and then solve the equation efficiently. First, the singular problem is transformed to a Fredholm integral equation of the second kind via Green’s function. Second, the truncated fractional series of the solution about the singularity is formulated by using Picard iteration and implementing series expansion for the nonlinear function. Third, a suitable variable transformation is performed by using the known singular information of the solution such that the solution of the transformed equation is sufficiently smooth. Fourth, the Chebyshev collocation method is used to solve the deduced equation to obtain approximate solution with high precision. Fifth, the convergence analysis of the collocation method is conducted in weighted Sobolev spaces for linear singular equations. Sixth, numerical examples confirm the effectiveness of the algorithm. Finally, the Thomas–Fermi equation and the Emden–Fowler equation as some applications are accurately solved by the method.

On multiple solutions for a fourth order nonlinear singular boundary value problems arising in epitaxial growth theory

In this article, we consider the fourth order non‐self‐adjoint singular boundary value problem with λ as a parameter measures the speed at which new particles are deposited. This differential equation is non‐self‐adjoint, so finding its solutions is not easy by usual methods. Also, this does not have a unique solution; therefore, finite differences and discrete methods may not be applicable. Here, to find approximate solutions, we propose to use the Adomian decomposition method (ADM) and compute the approximate solutions which are fully dependent on the size of the parameter λ. Here, we prove theoretically that the approximate solutions converge to the exact solutions. For small positive values of parameter λ, the above equation has two solutions. No solutions could be found for large positive values of the parameter. The uniqueness of the solution may not be guaranteed for a fixed positive value of the parameter. For negative values of parameter existences of the solution are always guaranteed.

An efficient numerical method based on exponential B‐spline basis functions for solving a class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions

In this paper, we develop a numerical scheme to approximate the solution of a general class of nonlinear singular boundary value problems (SBVPs) subject to Neumann and Robin boundary conditions. The original differential equation has a singularity at the point which is removed via L'Hospital's law with an assumption about the derivative of the solution at the point An exponential B‐spline collocation approach is then constructed to solve the resulting boundary value problem. Convergence analysis of the method is discussed. Numerical examples are provided to illustrate the applicability and efficiency of the method. Our results are compared with those obtained by other three numerical methods such as uniform mesh cubic B‐spline collocation (UCS) method, nonstandard finite difference method, and finite difference method based on Chawla's identity.

Blow-Up Solutions for a Singular Nonlinear Hadamard Fractional Boundary Value Problem

We consider singular nonlinear Hadamard fractional boundary value problems. Using properties of Green’s function and a fixed point theorem, we show that the problem has positive solutions which blow up. Finally, some examples are provided to explain the applications of the results.

ON THE DIRICHLET PROBLEM FOR AN ELLIP-TIC FUNCTIONAL-DIFFERENTIAL EQUATION WITH AFFINE TRANSFORMATION OF THE ARGUMENT

The paper considers the Dirichlet problem for an elliptic functionaldifferential equation containing a combination of shifts and contractions of the argument of an unknown function under the Laplace operator. Sufficient conditions for unique solvability are established. It is also shown that the problem can have an infinite-dimensional variety of solutions. This work is done in the framework of the State Contract in accordance with the Additional Agreement No. 075-03-2020-239/2 from 07.07.2020, registry No. 248, Budgetary Classification Code 01104730290059611, for the project titled ‘Non-linear singular integro-differential equations and boundary value problems’. The authors also acknoowledge support by the Russian Foundation for Basic Research (project no. 18-41-200001p-a).

Existence and nonexistence results for a class of fourth‐order coupled singular boundary value problems arising in the theory of epitaxial growth

In this article, we propose novel coupled nonlinear singular boundary value problems arising in epitaxial growth theory. The coupled equations are nonlinear, non‐self‐adjoint, and singular and have no exact solutions. We derive some qualitative properties of the coupled solutions, which depend on the size of parameters that occur in the coupled system. To prove the existence of the coupled solutions, we apply the monotone iterative technique on equivalent coupled integral equations in the presence of upper and lower solutions. We also compute the bounds for the parameters, which indicates the existence of coupled solutions for small values of the parameters, and nonexistence for large positive values of these parameters. To illustrate the theory developed, we consider test cases and compute the sequence of upper and lower solutions. To verify the bounds on the parameters, we have used a type of Adomian decomposition method.

Existence and multiplicity of positive solutions for a new class of singular higher-order fractional differential equations with Riemann\u2013Stieltjes integral boundary value conditions

In this work, the aim is to discuss a new class of singular nonlinear higher-order fractional boundary value problems involving multiple Riemann–Liouville fractional derivatives. The boundary conditions are constituted by Riemann–Stieltjes integral boundary conditions. The existence and multiplicity of positive solutions are derived via employing the Guo–Krasnosel’skii fixed point theorem. In addition, the main results are demonstrated by some examples to show their validity.

Three-Order Multipoint Boundary Value Problems for p-Laplacian Operator on Time Scales

In this paper, the existence of positive solutions for the nonlinear four-point singular BVP for there-order with p-Laplacian operator on time scales will be studied. By using the fixed-point theory, the existence of positive solutions for nonlinear singular boundary value problem with p-Laplacian operator on time scales is obtained.

A Review on a Class of Second Order Nonlinear Singular BVPs

Several real-life problems are modeled by nonlinear singular differential equations. In this article, we study a class of nonlinear singular differential equations, explore its various aspects, and provide a detailed literature survey. Nonlinear singular differential equations are not easy to solve and their exact solution does not exist in most cases. Since the exact solution does not exist, it is natural to look for the existence of the analytical solution and numerical solution. In this survey, we focus on both aspects of nonlinear singular boundary value problems (SBVPs) and cover different analytical and numerical techniques which are developed to deal with a class of nonlinear singular differential equations − ( p ( x ) y ′ ( x ) ) ′ = q ( x ) f ( x , y , p y ′ ) for x ∈ ( 0 , b ) , subject to suitable initial and boundary conditions. The monotone iterative technique has also been briefed as it gained a lot of attention during the last two decades and it has been merged with most of the other existing techniques. A list of SBVPs is also provided which will be of great help to researchers working in this area.

A fourth-order non-uniform mesh optimal B-spline collocation method for solving a strongly nonlinear singular boundary value problem describing electrohydrodynamic flow of a fluid

In this paper, a non-uniform mesh optimal B-spline collocation method is presented for the numerical solution of a singular two-point boundary value problem describing electrohydrodynamic flow (EF) of a fluid in a circular cylindrical conduit. The EF problem is highly nonlinear and has a singularity at the point r=0. Further, this problem has a boundary layer near right end of the solution domain. We consider a grading function to construct a non-uniform mesh over the problem domain. The non-uniform mesh is constructed in such a way that the mesh is finer near the right end boundary. Convergence of the proposed method is analyzed. The EF problem is solved for small and large values of the two relevant parameters: (i) strength of non-linearity β, and (ii) Hartmann electric number H. The effects of β and H on the velocity field are investigated. The computed results have been compared with those obtained by two other B-spline collocation methods over uniform mesh to show the advantage of the present method. It is shown that the present method provides O(h4) convergent approximation.

Numerical solution of singular boundary value problems using advanced Adomian decomposition method

In this work, we present a powerful method for the numerical solution of non-linear singular boundary value problems, namely the advanced Adomian decomposition method which is the modification of the Adomian decomposition method. In this method, we use all the boundary conditions to obtain the coefficient of the approximate series solution. Moreover, convergence analysis and an error bound for the approximate solution are discussed. This method overcomes the singular behaviour of problems and illustrates the approximations of high precision with a large effective region of convergence. To prove the robustness and effectiveness of the proposed method, various examples are considered and the obtained results are compared with the other existing methods.

Variable Mesh Polynomial Spline Discretization for Solving Higher Order Nonlinear Singular Boundary Value Problems

In this paper, two generalized variable mesh finite difference schemes based on cubic spline has been developed to solve the system of nonlinear singular boundary value problems. The suggested methods are pertinent to singular boundary value problem and are of second and third order. Numerical examples are provided to prove the precision and competence of the schemes.

A note on the convergence of fuzzy transformed finite difference methods

In this paper, we develop numerical methods based on the fuzzy transform methods (FTMs). In this approach we apply fuzzy transforms on discrete version of the derivatives and use it to derive FTMs. We also establish convergence of the proposed FTMs. To test the efficiency of the proposed FTMs, we apply the FTM schemes on the second order nonlinear singular boundary value problems and fourth order BVPs. We allow the source term of the differential equation to have jump discontinuity and study the effect of jump on FTMs and finite difference methods. The work shows that FTMs are better for both class of BVPs considered in this paper, having singularity, nonlinearity and jump discontinuity.

Evolutionary numerical approach for solving nonlinear singular periodic boundary value problems

In this approximation study, a nonlinear singular periodic model in nuclear physics is solved by using the Hermite wavelets (HW) technique coupled with a numerical iteration technique such as the Newton Raphson (NR) one for solving the resulting nonlinear system. The stimulation of offering this numerical work comes from the aim of introducing a consistent framework that has as effective structures as Hermite wavelets. Two numerical examples of the singular periodic model in nuclear physics have been investigated to observe the robustness, proficiency, and stability of the designed scheme. The proposed outcomes of the HW technique are compared with available numerical solutions that established fitness of the designed procedure through performance evaluated on a multiple execution.

Iterative unique positive solutions for a new class of nonlinear singular higher order fractional differential equations with mixed-type boundary value conditions

In this paper, we consider a new class of singular nonlinear higher order fractional boundary value problems supplemented with sum of Riemann–Stieltjes integral type and nonlocal infinite-point discrete type boundary conditions. The fractional derivative of different orders is involved in the nonlinear terms and boundary conditions, and the nonlinear terms are allowed to be singular in regard to not only time variable but also space variables. A unique positive solution is established by using the fixed point theorem of mixed monotone operator. In addition, some significant properties of the unique solution depending on the parameter λ are stated. In the end, two examples are worked out to illustrate our main results.