Singular Equations Research Articles

A New Blow-Up Criterion to a Singular Non-Newton Polytropic Filtration Equation

In this paper, a singular non-Newton polytropic filtration equation under the initial-boundary value condition is revisited. The finite time blow-up results were discussed when the initial energy E(u0) was subcritical (E(u0)<d), critical (E(u0)=d), and supercritical (E(u0)>d), with d being the potential depth by using the potential well method and some differential inequalities. The goal of this paper is to give a finite time blow-up result if E(u0) is independent of d. Moreover, the explicit upper bound of the blow-up time is obtained by the classical Levine’s concavity method, and the precise lower bound of the blow-up time is derived by applying an interpolation inequality.

Artificial neural network for solving the nonlinear singular fractional differential equations

This paper proposes an artificial neural network (ANN) architecture for solving nonlinear fractional differential equations. The proposed ANN algorithm is based on a truncated power series expansion to substitute the unknown functions in the equations in this approach. Then, a set of algebraic equations is resolved using the ANN technique in an iterative minimization process. Finally, numerical examples are provided to demonstrate the usefulness of the ANN architectures. The results verify that the suggested ANN architecture achieves high accuracy and good stability.

Numerical Solutions for Singular Lane-Emden Equations Using Shifted Chebyshev Polynomials of the First Kind

This paper describes an algorithm for obtaining approximate solutions to a variety of well-known Lane-Emden type equations. The algorithm expands the desired solution y(x) ≃ yN(x), in terms of shifted Chebyshev polynomials of first kind such that yN(i)(0) = y(i)(0) (i = 0, 1, ..., N). The derivative values y(j)(0) for j = 2, 3, ..., are computed by using the given differential equation and its initial conditions. This makes approximate solutions more consistent with the exact solutions of given differential equations. The explicit expressions of the expansion coefficients of yN(x) are obtained. The suggested method is much simpler compared to any other method for solving this initial value problem. An excellent agreement between the exact and the approximate solutions is found in the given examples. In addition, the error analysis is presented.

EXISTENCE, UNIQUENESS AND REGULARITY OF THE SOLUTION FOR A SYSTEM OF WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND

The main purpose of this paper is to investigate the existence, uniqueness, and regularity of solutions for systems of linear and nonlinear weakly singular Volterra integral equations of the first kind, which have been stated as research problems (for the linear case) by Brunner (2017). To get these results, we first state sufficient conditions for the existence and uniqueness of solutions for a system of weakly singular Volterra integral equations of the second kind. We demonstrate the results by giving an example.

On a spectral theory of singular Hahn difference equation of a Sturm‐Liouville type problem with transmission conditions

This study is based upon investigations of the existence of a singular Hahn difference equation of ‐Sturm–Liouville problem with transmission conditions. Moreover, the Parseval identity and expansion formula in eigenfunctions are obtained.

A Systematic Review on the Solution Methodology of Singularly Perturbed Differential Difference Equations

This review paper contains computational methods or solution methodologies for singularly perturbed differential difference equations with negative and/or positive shifts in a spatial variable. This survey limits its coverage to singular perturbation equations arising in the modeling of neuronal activity and the methods developed by numerous researchers between 2012 and 2022. The review covered singularly perturbed ordinary delay differential equations with small or large negative shift(s), singularly perturbed ordinary differential–differential equations with mixed shift(s), singularly perturbed delay partial differential equations with small or large negative shift(s) and singularly perturbed partial differential–difference equations of the mixed type. The main aim of this review is to find out what numerical and asymptotic methods were developed in the last ten years to solve such problems. Further, it aims to stimulate researchers to develop new robust methods for solving families of the problems under consideration.

On the convergence of piecewise polynomial collocation methods for variable-order space-fractional diffusion equations

In this study, we present a piecewise polynomial collocation method for the boundary-value problem of variable-order linear space-fractional diffusion equations. The proposed model is transformed to a weakly singular Volterra integral equation (VIE) of the second kind by an auxiliary variable, then a collocation method is constructed and analyzed for the obtained VIE. We demonstrate the existence and uniqueness of the collocation solution, as well as the optimal convergence order of the collocation method. Some numerical experiments are given to illustrate the theoretical results.

NEW DEVELOPMENT OF ADOMIAN DECOMPOSITION METHOD FOR SOLVING SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

This study attempts at Adomian Decomposition Method (ADM) for solving second order ordinary differential equations such as Multi singular equation, Bessel’s equation, and Oscillatory systems. The (ADM) results in many known equations, some of them are studied and investigated by mathematicians and researchers, while others are still not well researched yet. We give a new differential operator and invers differential operator to solve different types for initial value problem in the second order ordinary differential equation. There are some different situations that we will study and give the non-linear examples that lead us to the approximate solutions of exact solution.

Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed.

On Approximate Solution of One Class of Singular Integro-Differential Equations

A linear integro-differential equation with a singular differential operator in the principal part is studied. For its approximate solution in the space of generalized functions, special generalized versions of the methods of moments and subdomains are proposed and substantiated. Optimality of the methods in order of accuracy is established.

On Approximate Solution of One Class of Singular Integro-Differential Equations

On Approximate Solution of One Class of Singular Integro-Differential Equations

On the fractional Kirchhoff equation with critical Sobolev exponent

We provide four existence results of positive solutions to the fractional singular perturbation Kirchhoff equation using the finite-dimensional reduction approach and perturbed argument.

MAXIMUM PRINCIPLE FOR SINGULAR CONTROL PROBLEMS OF SYSTEMS DRIVEN BY MARTINGALE MEASURES

We provide necessary optimality conditions for singular controlled stochastic differential equations driven by an orthogonal continuous martingale measure. The control is allowed to enter both the drift and diffusion coefficient and has two components, the first being relaxed and the second singular, the domain of the first control does not need to be convex, and for the relaxing method, we show by a counter-example that replacing the drift and diffusion coefficients by their relaxed counterparts does not define a true relaxed control problem. The maximum principle for these systems is established by means of spike variation techniques on the relaxed part of the control and a convex perturbation on the singular one. Our result is a generalization of Peng's maximum principle to singular control problems.

Boundary regularity for viscosity solutions of nonlinear singular elliptic equations

Boundary regularity for viscosity solutions of nonlinear singular elliptic equations

The structure group for quasi-linear equations via universal enveloping algebras

We replace trees by multi-indices as an index set of the abstract model space to tackle quasi-linear singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group, which arises from a Hopf algebra and a comodule. Our approach, where the dual of the abstract model space naturally embeds into a formal power series algebra, allows to interpret the structure group as a Lie group arising from a Lie algebra consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (non-linearities, functions of space-time mod constants). We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the stochastic heat equation.

Oblique derivative problem in a plane sector for strong quasi-linear elliptic equation with p-Laplacian

ABSTRACT We study the behavior near the boundary corner point of weak bounded solutions to the oblique derivative problem for singular p-Laplacian equation with strong nonlinearities singular absorption term in a bounded plane sector. However, no work has been done for investigating the behavior of solutions to the oblique problem for singular p-Laplacian equation in a corner.

A Lagrange spectral collocation method for weakly singular fuzzy fractional Volterra integro-differential equations

A Lagrange spectral collocation method for weakly singular fuzzy fractional Volterra integro-differential equations

Positive solutions to integral boundary value problems for singular delay fractional differential equations

<abstract><p>Delay fractional differential equations play very important roles in mathematical modeling of real-life problems in a wide variety of scientific and engineering applications. The objective of this manuscript is to study the existence and uniqueness of positive solutions for singular delay fractional differential equations with integral boundary data. To investigate the described system, we construct a $ u_0 $-positive operator first. New research technique of by constructing $ u_0 $-positive operator is used to overcome the difficulties caused by both the delays and the boundary value conditions. Then the sufficient conditions for the existence and uniqueness of positive solutions of a class of the singular delay fractional differential equations with integral boundary is proved by using the fixed point theorem in cone.</p></abstract>

Extension of linear operators with application

The article presents a method for solving characteristic singular integral equations perturbed with compact operators. The method consists in reducing the solution of these equations to the solution of the systems of singular (unperturbed) equations, which are solved by factoring the coefficients of the obtained systems. The method presented concerns some results of Gohberg and Krupnik and can be used in solving other classes of functional equations with composite operators that fit into the scheme described by Theorem 1.1.

Existence and stability results for nonlinear coupled singular fractional-order differential equations with time delay

<abstract><p>The objective of the manuscript is to build coupled singular fractional-order differential equations with time delay. To study the underline problem, an integral representation is initially discussed and the operator form of the solution is investigated using various supplementary hypotheses. Also, the existence and uniqueness of the considered problem are investigated by using the Lebesgue-dominated convergence theorem and some analysis results. Moreover, the stability analysis to determine the nature of the proposed model's solution is examined. Finally, two supportive examples are provided to demonstrate our analysis as applications.</p></abstract>