Nonlinear Singular Equations Research Articles

Continuous Dependence of the Singular Nonlinear Van der Pol Equation Solutions with Respect to the Boundary Conditions: Elements of p-regularity Theory

This paper studies the problem of the continuous dependence of Van der Pol equation solutions with respect to the boundary conditions. We provide a new approach for the existence of such solutions via p-regularity theory. Several existence theorems about continuous solutions are established.

Global existence and decay for a system of two singular one-dimensional nonlinear viscoelastic equations with general source terms

ABSTRACT The paper studies a system of two singular one-dimensional nonlinear equations that arise in generalized viscoelasticity with long-term memory, with general source terms and nonlocal boundary condition. We prove the existence of a global solution to the problem using the potential-well theory. Furthermore, we construct a Lyapunov functional and use it together with the perturbed energy method to prove a general decay result.

The vanishing exponent limit for motion by a power of mean curvature

We discuss limit behavior of solutions to level set equation of power mean curvature flow as the exponent tends to zero, which has important applications to shape analysis in image processing. A formal limit yields a fully nonlinear singular equation that describes the motion of a surface by the sign of its mean curvature.We justify the convergence by providing a definition of viscosity solutions to the limit equation and establishing a comparison principle.

A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions

The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.

A study of fracture of a fibrous composite

We develop design model within which nucleation and propagation of crack in a fibrous composite is described. It is assumed that under loading, crack initiation and fracture of material happens in the composite. The problem of equilibrium of a composite with embryonic crack is reduced to the solution of the system of nonlinear singular integral equations with the Cauchy type kernel. Normal and tangential forces in the crack nucleation zone are determined from the solution of this system of equations. The crack appearance conditions in the composite are formed with regard to criterion of ultimate stretching of the material\'s bonds. We study the case when near the fiber, the binder has several arbitrary arranged rectilinear prefracture zones and a crack with interfacial bonds. The proposed computational model allows one to obtain the size and location of the zones of damages (prefracture zones) depending on geometric and mechanical characteristics of the fibrous composite and applied external load. Based on the suggested design model that takes into account the existence of damages (the zones of weakened interparticle bonds of the material) and cracks with end zones in the composite, we worked out a method for calculating the parameters of the composite, at which crack nucleation and crack growth occurs.

On Some Initial and Initial Boundary Value Problems for Linear and Nonlinear Boussinesq Models

The main concern of this paper is to apply the modified double Laplace decomposition method to a singular generalized modified linear Boussinesq equation and to a singular nonlinear Boussinesq equation. An a priori estimate for the solution is also derived. Some examples are given to validate and illustrate the method.

A Fractional Spectral Collocation for Solving Second Kind Nonlinear Volterra Integral Equations with Weakly Singular Kernels

The classical integer-order Jacobi spectral methods for solving second kind nonlinear Volterra integral equations with weakly singular kernels may cause a low-order accuracy in numerically approximating the exact solution. To overcome the shortcomings, we in this paper present a fractional spectral collocation method for solving weakly singular nonlinear Volterra integral equations. Based on the behavior of the original solution near the initial point of integration, we construct the fractional interpolation basis in the collocation method, and then develop an easily implementing technique to approximate the entry with one-fold integral in the resulting nonlinear system produced by the fractional spectral method. Consequently, we establish that both the semi-discrete and the fully discrete nonlinear systems have a unique solution for sufficiently large n, respectively, where $$n+1$$ denotes the dimension of the approximate space. We also ensure that two approximate solutions produced by both the semi-discrete and the fully discrete method arrive at the quasi-optimal convergence order in the infinite norm. At last, numerical examples are given to confirm the theoretical results.

A UNIQUENESS THEOREM FOR A NONLINEAR SINGULAR INTEGRAL EQUATION ARISING IN $ p $-ADIC STRING THEORY

We study a singular nonlinear integral equation on the real line that appear in $ p $-adic string theory. A uniqueness theorem for this equation in certain class of odd functions is proved. At the end of the paper we give examples, satisfying the conditions of the formulated theorem.

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _p u = H(u)\mu &{}\quad \text {in}\ \Omega ,\\ u>0 &{}\quad \text {in}\ \Omega ,\\ u=0 &{}\quad \text {on}\ \partial \Omega . \end{array}\right. } \end{aligned}$$Here \(\Omega \) is an open bounded subset of \({\mathbb {R}}^N\) (\(N\ge 2\)), \(\Delta _p u:= {\text {div}}(|\nabla u|^{p-2}\nabla u)\) (\(1<p<N\)) is the p-laplacian operator, \(\mu \) is a nonnegative bounded Radon measure on \(\Omega \) and H(s) is a continuous, positive and finite function outside the origin which grows at most as \(s^{-\gamma }\), with \(\gamma \ge 0\), near zero.

Explicit criteria for exponential stability of nonlinear singular equations with delays

By a novel approach, we give some explicit criteria for global exponential stability of singular nonlinear differential equations with delays. An application to electrical networks containing lossless transmission lines is presented.

Solution of Initial and Boundary Value Problems by an Effective Accurate Method

The newly proposed analytic approximate solution method in the recent publications [Turkyilmazoglu, M. [2013] “Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type,” Appl. Math. Mod. 37, 7539–7548; Turkyilmazoglu, M. [2014] “An effective approach for numerical solutions of high-order Fredholm integro-differential equations,” Appl. Math. Comput. 227, 384–398; Turkyilmazoglu, M. [2015] “Parabolic partial differential equations with nonlocal initial and boundary values,” Int. J. Comput. Methods, doi: 10.1142/S0219876215500243] is extended in this paper to solve initial and boundary value problems governed by any order linear differential equations whose exact solutions are hard to obtain. Exact solutions are found from the method when the solutions are themselves polynomials. Better accuracies are achieved within the method by increasing the number of polynomials. Comparisons with some available methods show the ability of the proposed technique, even performing much better than the traditional Taylor series expansion.

Dirichlet Boundary Conditions for Degenerate and Singular Nonlinear Parabolic Equations

We study existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations, in bounded domains. We show that there exists a unique solution which satisfies possibly inhomogeneous Dirichlet boundary conditions. To this purpose some barrier functions are properly introduced and used.

Numerical Solution of Nonlinear Mixed Integral Equation with a Generalized Cauchy Kernel

In this article, we present approximate solution of the two-dimensional singular nonlinear mixed Volterra-Fredholm integral equations (V-FIE), which is deduced by using new strategy (combined Laplace homotopy perturbation method (LHPM)). Here we consider the V-FIE with Cauchy kernel. Solved examples illustrate that the proposed strategy is powerful, effective and very simple.

A spectral iterative method for solving nonlinear singular Volterra integral equations of Abel type

In this paper, a spectral iterative method is employed to obtain approximate solutions of singular nonlinear Volterra integral equations, called Abel type of Volterra integral equations. The Abel's type nonlinear Volterra integral equations are reduced to nonlinear fractional differential equations. This approach is based on a combination of two different methods, i.e. the iterative method proposed in [7] and the spectral method. The method reduces the fractional differential equations to systems of linear algebraic equations and then the resulting systems are solved by a numerical method. Finally, we prove that the spectral iterative method (SIM) is convergent. Numerical results comparing this iterative approach with alternative approaches offered in [4,8,24] are presented. Error estimation also corroborate numerically.

Approximate bond graph models for linear singularly perturbed systems

ABSTRACTA method for obtaining approximate bond graph models for linear time invariant (LTI) Multi-Input Multi-Output (MIMO) systems with singular perturbations is presented. The basic idea of using time-scale analysis in obtaining low-order models is to decouple the slow and fast models. This is achieved by using two-stage linear transformations. Hence, a procedure to construct decoupled bond graph models based on -fields representing each dynamic of the singularly perturbed system is proposed.When the linear transformations are applied to the system with singular perturbations, non-linear and linear equations have to be solved for separating the subsystems. In many cases, the exact solutions of these equations are complicated, but approximate solutions can be determined and approximate models can be obtained.Thus, zeroth- and first-order solutions in a bond graph approach are proposed. The key to finding the approximate solutions is to obtain the relations of the original bond graph with a predefined integral causality of the system and another bond graph called the Singularly Perturbed Bond Graph whose storage elements of the fast dynamics have derivative causality and for the slow dynamics they maintain an integral causality assignment.Finally, the proposed method is applied to an illustrative example where the simulation results show the exact solutions and zeroth- and first-order approximations.

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.

Solving the Two–Dimensional Lane–Emden Type Equations by the Adomian Decomposition Method

In this paper, we introduce the two–dimensional Lane–Emden type equations. We study the linear and the nonlinear forms of these types of equations. We use the systematic Adomian decomposition method to handle these equations with specified initial conditions. The Adomian decomposition method provides an efficient scheme for exact and approximate analytic solutions of singular nonlinear equations. We corroborate this work by studying several two-dimensional Lane-Emden problems that represent numerical examples to show the strength of our approach.

Nagumo theorems of third-order singular nonlinear boundary value problems

In this paper, we establish the Nagumo theorems for boundary value problems associated with a class of third-order singular nonlinear equations: $(p(t)x')''=f(t,x,p(t)x',(p(t)x')')$ , $\forall t\in(0,1)$ by the method of upper and lower solutions and the Schauder fixed point theorem. We also consider the multiplicity of the solutions by using topological degree theory. There are some examples to illustrate how the results of this paper can be applied.

Optimal control of singular stationary systems with phase constraints and state variation

An optimal control problem for a system described by a singular nonlinear equation of elliptic type with an inclusion phase constraint is considered. Necessary optimality conditions are obtained by varying system states.

On the multi-point Levenberg–Marquardt method for singular nonlinear equations

In this paper, we present a multi-point iterative Levenberg–Marquardt algorithm for singular nonlinear equations. The algorithm converges globally and the convergence order is studied under the local error bound condition, which is weaker than the nonsingularity of the Jacobian at the solution.