Nonlinear Singular Problems Research Articles

A novel Chebyshev neural network approach for solving singular arbitrary order Lane-Emden equation arising in astrophysics

ABSTRACT The motivation of this investigation is to develop a single-layer Chebyshev Neural Network (ChNN) model to handle singular fractional (arbitrary)-order Lane-Emden type equations. These equations are well-known application problems of astrophysics and quantum mechanics. Fractional Lane-Emden equations are singular so it is very difficult to solve analytically. Thus, an efficient method is required to handle the above equations. Here, our main aim is to use a single-layer ChNN model for solving fractional Lane-Emden equations. ChNN model is one kind of Functional Link Neural Network (FLNN) in which the hidden layer is replaced by a functional expansion block of the input pattern using orthogonalshifted Chebyshev polynomials (SChP). Thus, the network parameters of ChNN are less than the Multi-Layer Artificial Neural Network (MLANN). We have considered factional-order singular nonlinear problems of astrophysics to show the computational effort of the proposed method. Back Propagation algorithm of the unsupervised version has been considered for minimizing the error function and updating the weights of the ChNN model. Computed results are displayed in terms of tables and graphs.

Nonlinear singular problems with indefinite potential and a superlinear perturbation

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian plus an indefinite potential. In the reaction, we have the competing effects of a parametric singular term and of a superlinear perturbation. We examine the nonexistence, existence and multiplicity of positive solutions as the parameter varies.

Haar wavelet quasilinearization method for numerical solution of Emden–Fowler type equations

In this paper, an efficient method for solving the nonlinear Emden–Fowler type boundary value problems with Dirichlet and Robin–Neumann boundary conditions is introduced. The present method is based on the Haar-wavelets and quasilinearization technique. The quasilinearization technique is adopted to linearize the nonlinear singular problem. Numerical solution of linear singular problem is obtained by the Haar wavelet method. The numerical study is further supported by examining thoroughly the convergence of the Haar wavelet method and the quasilinearization technique. In order to check the accuracy of the proposed method, the numerical results are compared with both existing methods and exact solutions.

Nonlinear singular problems with indefinite potential term

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term will be parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter lambda varies.

On some singular nonlinear problems for monotone elliptic operators

The goal of this note is to study some class of problems associated to nonlinear operators of the p -Laplacian type with source term having a singularity at the origin.

Continuity results for parametric nonlinear singular Dirichlet problems

AbstractIn this paper we study from a qualitative point of view the nonlinear singular Dirichlet problem depending on a parameterλ> 0 that was considered in [32]. Denoting bySλthe set of positive solutions of the problem corresponding to the parameterλ, we establish the following essential properties ofSλ:there exists a smallest element$\begin{array}{} u_\lambda^* \end{array}$inSλ, and the mappingλ↦$\begin{array}{} u_\lambda^* \end{array}$is (strictly) increasing and left continuous;the set-valued mappingλ↦Sλis sequentially continuous.

Compact half step approximation in exponential form for the system of 2D second-order quasi-linear elliptic partial differential equations

ABSTRACTThis paper deals with devising a novel exponential scheme, implicit in nature, using half step discretization of order four for computing the numerical solution of quasi-linear elliptic partial differential equations of the type , . Nine mesh points in a single compact cell are being used in this method. An elaborate convergence analysis of the proposed method has been worked out. The technique for a scalar equation is eventually extended to solve the systems of quasi-linear elliptic equations. The technique is then applied to noteworthy problems and computational outcomes corroborate the theoretical rate of convergence. Linear and non-linear singular problems are tackled separately ensuring the usage of nine spatial grid points of a single computing cell.

A Modified Homotopy Perturbation Method for Nonlinear Singular Lane–Emden Equations Arising in Various Physical Models

A modified homotopy perturbation method for solving a class of nonlinear Lane–Emden equations with boundary conditions arising in various physical models is proposed. The proposed algorithm is based on the homotopy perturbation method and integral form of the Lane–Emden equation. The integral form of the problem overcomes the singular behavior at the origin. The accuracy and applicability of our algorithm is examined by solving two singular models: (i) the second kind Lane–Emden equation used to model a thermal explosion in an infinite cylinder or a sphere and (ii) the nonlinear singular problem with Neumann boundary conditions.

Existence and Global Asymptotic Behavior of Singular Positive Solutions for Radial Laplacian

The aim of this paper is to establish existence and uniqueness of a positive continuous solution to the following singular nonlinear problem. {-t1-ntn-1u′′=a(t)uσ, t∈(0,1), limt→0⁡tn-1u′(t)=0, u(1)=0}, where n≥3,σ<1, and a denotes a nonnegative continuous function that might have the property of being singular at t=0 and /or t=1 and which satisfies certain condition associated to Karamata class. We emphasize that the nonlinearity might also be singular at u=0, while the solution could blow-up at 0. Our method is based on the global estimates of potential functions and the Schauder fixed point theorem.

High-order compact difference scheme of 1D nonlinear degenerate convection–reaction–diffusion equation with adaptive algorithm

This research describes a high accuracy adaptive difference strategy on 1D reaction–diffusion equation with convection item, which explains quenching phenomena of nonlinear singular degenerate problem. It originates from the first and the second central difference scheme and the forward difference operator. The former is used to discrete the first-order and the second-order spatial derivative and the latter is utilized to approximate temporal derivative. After its accuracy and stability discussed, the typical quenching cases are illustrated singularity of the nonlinear problem by using the proposed strategy with adaptation mechanism and comparing with the low-order approach.

Application of a coupled approach for the solution of nonlinear singular initial value problems of Lane–Emden type

Lane–Emden type equations have been interesting since long time due to their wide applications in mathematical physics and astrophysics. In this paper, a coupled approach has been proposed for the solution of nonlinear singular value problems of Lane–Emden type.

A multiplicity results for a singular problem involving a Riemann-Liouville fractional derivative

In this work, we investigate the following nonlinear singular problem with Riemann-Liouville Fractional Derivative (P?) {-tD?1(?0D?t (u(t))?p-2 0D? tu(t)) = g(t)/u?(t) + ?f (t,u(t)) t ? (0,T); u(0) = u(T) = 0, where ? is a positive parameter, p > 1, 1/2 < ? ? 1,0 < ? < 1, g ? C([0,T]) and f?C([0,T] x R,R). Under appropriate assumptions on the function f, we employ the method of the Nehari manifold combined with the fibering maps in order to show the existence of ?0 such that for all ? ?(0,?0) the problem (P?) has at least two positive solutions. Finally, some examples are given to illustrate our results.

A nonlinear singular eigenvalue problem for a linear system of ordinary differential equations with redundant conditions

A nonlinear eigenvalue problem for a linear system of ordinary differential equations is examined on a semi-infinite interval. The problem is supplemented by nonlocal conditions specified by a Stieltjes integral. At infinity, the solution must be bounded. In addition to these basic conditions, the solution must satisfy certain redundant conditions, which are also nonlocal. A numerically stable method for solving such a singular overdetermined eigenvalue problem is proposed and analyzed. The essence of the method is that this overdetermined problem is replaced by an auxiliary problem consistent with all the above conditions.

Iterative method for solving nonlinear singular problems

The aim of our work is to present a method (p-factor method) for solving nonlinear equations of the form $$\begin{aligned} F(x)=0, \quad F:\mathbb {R}^n\rightarrow \mathbb {R}^m, \end{aligned}$$F(x)=0,F:RnźRm,in singular (irregular) case, i.e. when the matrix $$F'(x)$$Fź(x) is singular at an initial point $$x_0$$x0 of an iterative sequence $$\{x_k\}$${xk}, $$k=1,2,\ldots $$k=1,2,ź We investigate conditions that have to be fulfilled at the initial point $$x_0$$x0 to obtain the existence of solution of nonlinear equation $$F(x)=0$$F(x)=0, prove convergence of the presented p-factor method and give estimation of convergence rate for this method.

A nonlinear singular eigenvalue problem for a Hamiltonian system of differential equations with redundant condition

A nonlinear eigenvalue problem for a self-adjoint Hamiltonian system of differential equations is examined on an infinite half-line. It is assumed that the original data (that is, the system matrix and the matrix of boundary conditions) satisfy certain monotonicity conditions for the spectral parameter. In addition to the initial condition and the requirement that the solution be bounded on infinity, a redundant nonlocal condition specified by a Stieltjes integral is imposed. In order to make the resulting problem nontrivially solvable, it is replaced by an auxiliary problem, which is consistent subject to all the above conditions. This auxiliary problem is examined, and a numerical method that solves it is given.

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English

Solution of nonlinear singular boundary value problems using polynomial-sinc approximation

A new highly accurate algorithm for the solution of nonlinear singular boundary value problems for ordinary differential equations is presented.The algorithm uses a collocation technique based on polynomial approximation at Sinc points. The scheme is tested for some nonlinear singular boundaryvalue problems showing an exponential convergence rate. The examples areof second and higher order singular, nonlinear boundary value problems. Foreach example the error formula of the approximation is discussed and verifiledin a comparison of the analytic solution

Existence and Uniqueness of Positive Solutions for Fourth-Order Nonlinear Singular Sturm-Liouville Problems

By mixed monotone method, we establish the existence and uniqueness of positive solutions for fourth-order nonlinear singular Sturm-Liouville problems. The theorems obtained are very general and complement previously known results.

Nonlinear singular elliptic problem with gradient term and general datum

In this paper we deal with an elliptic problem, with a first order term in natural growth, a singular nonlinearity and a L1 datum. We obtain an existence result without any smallness condition on the datum. We prove also a multiplicity result of positive solutions under additional conditions on the data.