Singular Differential Equations Research Articles

Polytropic Dynamical Systems with Time Singularity

In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method that combines two simple methods; Euler's method and shooting method, is proposed to approximate the solution of this type of dynamic equations. We adopt the shooting method to reduce the boundary value problem, then we apply Euler's algorithm to the resulted initial value problem to get approximations for the solution of the Lane-Emden equation. Finally, numerical examples and simulation are provided to show the validity and efficiency of the proposed technique, as well as the convergence and error estimation are analyzed.

Regular singular differential equations and free proalgebraic groups

AbstractWe determine the differential Galois group of the family of all regular singular differential equations on the Riemann sphere. It is the free proalgebraic group on a set of cardinality .

Optimal control of thermoregulation in the human dermal regions investigated through the stochastic integrated techniques

In this research, stochastic computing techniques based on artificial neural networks are applied to the proposed singular nonlinear differential equation to explain thermoregulation in the human dermal region problem to investigate and predict the effect of bioheat temperature on human skin at different atmospheric temperatures with different cases with considering the impact of both air convection thermal sensitivity and temperature, where sequential quadratic programming, interior point technique, and active set technique are included in the designed techniques. Moreover, thermal dynamics in the epidermal layer that is affected by both cooling and heating treatments, and the temperature profile across space and time have been examined. Numerical convergence analysis has been applied to endorse the nature of precision and convergence of the designed stochastic computing techniques. To preserve thermal balance, this leads to modifications in the human body's temperature regulation. The statistical analysis is provided comprehensively in the form of graphs and tables to further enhance the significance in terms of accuracy, efficiency, and convergence of the current study.

Numerical method for singular drift stochastic differential equation driven by fractional Brownian motion

In this paper, we study the stochastic differential equation with singular drift coefficient driven by fractional Brownian motion with Hurst parameter H∈(12,1). We obtain that the optimal convergence rate of the backward Euler method is 1.0. The constant elasticity of variance model and the Aït-Sahalia interest rate model are performed as numerical experiments to validate our theoretical results.

Existence and uniqueness of solution for a singular elliptic differential equation

Abstract In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: − Δ u − 1 2 ( x ⋅ ∇ u ) = μ h ( x ) u q − 1 + λ u − u p , x ∈ R N , u ( x ) → 0 , as ∣ x ∣ → + ∞ , \left\{\begin{array}{l}-\Delta u-\frac{1}{2}\left(x\cdot \nabla u)=\mu h\left(x){u}^{q-1}+\lambda u-{u}^{p},\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| x| \to +\infty ,\end{array}\right. where N ⩾ 3 N\geqslant 3 , 0 < q < 1 0\lt q\lt 1 , λ > 0 \lambda \gt 0 , p > 1 p\gt 1 , μ > 0 \mu \gt 0 is a parameter and the function h ( x ) h\left(x) satisfies certain conditions. To start with, based on the variational argument and perturbation method, we obtain the existence and uniqueness of the positive solution for the aforementioned singular elliptic differential equation as λ > N 2 \lambda \gt \frac{N}{2} . In addition, there is no solution as λ ⩽ N 2 \lambda \leqslant \frac{N}{2} . Later, from an experimental point of view, we give the numerical solution of the aforementioned singular elliptic differential equation by means of a neural network in some special cases, which enrich the theoretical results. Our conclusions partially extend the results corresponding to the nonsingular case.

Conformable Triple Sumudu Transform with Applications

One of the important topics in applied mathematics is the topic of integral transformations, due to their importance in electrical engineering applications, including communications in particular, and other sciences. In this work, one of the most important transformations in its three dimensions was presented, which is the triple Sumudu transform, including solving some real-life applications of physics, some of which have not been solved using such an integral transform before. In this work, we extend the Sumudu transform formula to the conformable fractional order, as well as other interesting and significant rules. The general analytical solution of a singular and nonlinear conformable fractional differential equation based on the conformable fractional Sumudu transform is also presented in this paper. The general solutions of several linear and nonhomogeneous conformable fractional differential equations can be obtained using the method we’ve proposed. As a result, our results reveal that our proposed method is an efficient one that can be used for solving all conformable fractional differential equations. The relationship between the Sumudu integral transform and other important and recently proposed integral transforms are also discussed. Finally, the triple Sumudu transform is used to solve boundary value problems, such as the heat equation with boundary values. The triple Sumudu integral transform is also used to solve linear partial integro-differential equations. The transform capability to handle such equations has been proven via its utilization in three applications.

Matkowski-Type Functional Contractions under Locally Transitive Binary Relations and Applications to Singular Fractional Differential Equations

This article presents a few fixed-point results under Matkowski-type functional contractive mapping using locally J-transitive binary relations. Our results strengthen, enhance, and consolidate numerous existent fixed-point results. To argue for the efficacy of our results, several illustrated examples are supplied. With the help of our findings, we deal with the existence and uniqueness theorems pertaining to the solution of a variety of singular fractional differential equations.

Singular Stochastic Differential Equations for Time Evolution of Stocks Within Non-white Noise Approach

Singular Stochastic Differential Equations for Time Evolution of Stocks Within Non-white Noise Approach

A study for a higher order Riemann-Liouville fractional differential equation with weakly singularity

<abstract><p>In this paper, we study an initial value problem with a weakly singular nonlinear fractional differential equation of higher order. First, we establish the existence of global solutions to the problem within the appropriate function space. We then introduce a generalized Riemann-Liouville mean value theorem. Using this theorem, we prove the Nagumo-type uniqueness theorem for the stated problem. Moreover, we give two examples to illustrate the applicability of the existence and uniqueness theorems.</p></abstract>

Multiplicity results for a class of nonlinear singular differential equation with a parameter

This paper gives sufficient conditions for the existence of positive periodic solutions to general indefinite singular differential equations. Furthermore, under some assumptions we show the existence of two positive periodic solutions. The methods used are Krasnoselski\(\breve{\mbox{i}}\)'s-Guo fixed point theorem and the positivity of the associated Green's function.

Existence and multiplicity of solutions for boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity

In this paper, we consider the existence of solutions for a boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity. By means of the Guo–Krasnosel’skii fixed point theorem and the Leray–Schauder nonlinear alternative theorem, we obtain some results on the existence and multiplicity of solutions, respectively.

The New Adomian Technique for Solving Singular and Non-singular Partial Differential Equations of First Order

In this paper, the proposed modification to Adomian decomposition method (ADM) is introduced. In addition, a new modification to solve singular and nonsingular initial-value problems (IVPs) in first order partial differential equations (PDEs) is investigated.
 Then, a discussion of how ADM can be used to solve singular and nonsingular IVPs is produced. Finally, some example are given to demonstrate the applicability of the results of paper.
 Key words; Adomian decomposition method; Nonlinear first-order PDE; Singular and nonsingular first-order partial differential equations.
 1 Introduction
 Singular nonlinear partial differential equations (PDEs) arise in various physical phe- nomena in applied sciences and engineering from such areas as fluid mechanics and heat-

A matrix technique‐based numerical treatment of a nonlocal singular boundary value problems

The mathematical modeling of the decisive event of astrophysics, physiology, and many other areas of science and technology witness the involvement of singular boundary value problems. The nonlocal boundary conditions are more informative than local boundary conditions (initial conditions and two‐point boundary conditions) to evaluate some mathematical models. This article presents a collocation approach‐based matrix technique to approximate the solution of the fusion of a class of singular differential equations subject to nonlocal three‐point boundary conditions. The proposed strategy utilizes the truncation of the series expansion of a function belonging to in terms of Bernoulli polynomials. It transforms the singular boundary value problems into a set of nonlinear algebraic equations, which can be dealt with by any mathematical software. The Lipschitz condition on an equivalent completely continuous nonlinear operator has been used to prove the convergence analysis of the scheme. Some extremely nonlinear test examples are solved and provided in contrast with the exact solution. These numerical results are also examined against some existing numerical techniques to verify the applicability and significance of the proposed methodology. There are a few numerical examples that are application based but do not have exact solutions. In such cases, residual error norm is employed to measure the accuracy of the numerical strategies. The computational data demonstrate the superiority and validity of the proposed technique over existing numerical approaches.

Biorthogonal flatlet multiwavelet collocation method for solving the singular nonlinear system with initial and boundary conditions

PurposeThe numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.Design/methodology/approachThe approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.FindingsAccording to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.Originality/valueThe key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

A Comparative Study of Numerical Solution of Second-order Singular Differential Equations Using Bernoulli Wavelet Techniques

The main objective of this article is to discuss a numerical method for solving singular differential equations based on wavelets. Singular differential equations are first transformed into a system of linear algebraic equations, and then the linear system’s solution produces the unknown coefficients. Along with its estimated error, the convergence of the approximative solution is alsodetermined. Some numerical examples are thought to show that Bernoulli wavelet is better than Chebyshev and Legendre wavelet and other existing techniques.

The g-Drazin inverses of anti-triangular block operator matrices

An element a in a Banach algebra A has g-Drazin inverse if there exists b∈A such that ab=ba,b=bab and a−a2b∈Aqnil. In this paper we find new explicit representations of the g-Drazin inverse of the block operator matrix (EIF0). We thereby solve a wider kind of singular differential equations posed by Campbell (1983) [2].

Systems of Singular Differential Equations as the Basis for Neural Network Modeling of Chaotic Processes

Systems of Singular Differential Equations as the Basis for Neural Network Modeling of Chaotic Processes

A Deep Learning Neural Network Framework for Solving Singular Nonlinear Ordinary Differential Equations

A Deep Learning Neural Network Framework for Solving Singular Nonlinear Ordinary Differential Equations

On $$(1+\sqrt{2})$$ convergence-based Newton–Raphson method for approximate solution of singular value Lane-Emden differential equation via modified Laplace Adomian decomposition technique

On $$(1+\sqrt{2})$$ convergence-based Newton–Raphson method for approximate solution of singular value Lane-Emden differential equation via modified Laplace Adomian decomposition technique

Solution of linear and nonlinear singular value problems using operational matrix of integration of Taylor wavelets

ABSTRACT In this paper, we present an effective method under Taylor wavelets and collocation technique to find an approximate solution of linear and non-linear second-order singular value differential equations. Application of this method on various problems confirms the efficiency, easy applicability, and rapid computation. The obtained solution is compared with some other existing numerical solutions. For many problems, this method gives a solution same as the exact solution of the problem which confirms the effectiveness and accuracy of this method. Additionally, we include graphs and figures to demonstrate that the Taylor wavelet method offers better accuracy for a number of problems. MATLAB software is used to process relative data and execute calculations.