Singular Differential Equations Research Articles

General-Kindred physics-informed neural network to the solutions of singularly perturbed differential equations

Physics-informed neural networks (PINNs) have become a promising research direction in the field of solving partial differential equations (PDEs). Dealing with singular perturbation problems continues to be a difficult challenge in the field of PINN. The solution of singular perturbation problems often exhibits sharp boundary layers and steep gradients, and traditional PINN cannot achieve approximation of boundary layers. In this manuscript, we propose the General-Kindred physics-informed neural network (GKPINN) for solving singular perturbation differential equations (SPDEs). This approach utilizes asymptotic analysis to acquire prior knowledge of the boundary layer from the equation and establishes a novel network to assist PINN in approximating the boundary layer. It is compared with traditional PINN by solving examples of one-dimensional, two-dimensional, and time-varying SPDE equations. The research findings underscore the exceptional performance of our novel approach, GKPINN, which delivers a remarkable enhancement in reducing the L2 error by two to four orders of magnitude compared to the established PINN methodology. This significant improvement is accompanied by a substantial acceleration in convergence rates, without compromising the high precision that is critical for our applications. Furthermore, GKPINN still performs well in extreme cases with perturbation parameters of 1×10−38, demonstrating its excellent generalization ability.

A new fourth-order compact finite difference method for solving Lane-Emden-Fowler type singular boundary value problems

We develop a novel fourth-order compact finite difference scheme to solve nonlinear singular ordinary differential equations. Such problems occur in many fields of science and engineering, such as studying the equilibrium of an isothermal gas sphere, reaction–diffusion in a spherical permeable catalyst, etc. These problems are challenging to solve because of their singularity or nonlinearity. By our proposed method, we can easily solve these complex problems without removing or modifying the singularity. To construct the new fourth-order compact difference method, Initially, we created a uniform mesh within the solution domain and developed a compact finite difference scheme. This scheme approximates the derivatives at the boundary nodal points to handle the problem’s singularity effectively. Employing a matrix analysis approach, we discussed the convergence analysis of the methods. To demonstrate its efficacy, we apply our approach to solve various real-life problems from the literature. The new method offers high-order accuracy with minimal grid points and provides better numerical results than the nonstandard finite difference method and exponential compact finite difference method.

Characteristics of solution to singular fractional differential equation with two Riemann-Stieltjesdintegral boundary value conditions

The purpose of this paper is to study unique solution and iterative sequence of approximate solution for uniformly approaching unique solution to a new class of singular fractional differential equations with two kinds of Riemann-Stieltjes integral boundary value conditions by using some fixed point theorems. Because of different properties of the nonlinear terms and complexity of the boundary conditions in equations, we first probe several fixed point theorems of sum-type operators which expand many existing works in this research area. It is essential to point out that some conditions in our works greatly simplify the proof process of fixed point theorems. By applying the operator conclusions obtained in this paper, some sufficient conditions that guarantee the existence and uniqueness of solutions to singular differential equations are obtained, two iterative schemes that uniformly converges to the unique solution are given which provide computational methods of approximating solutions. As applications, some examples are provided to illustrate our main results.

Existence of Positive Solutions for Singular Difference Equations with Nonlinear Boundary Conditions

In this paper, we delve into a discrete nonlinear singular semipositone problem, characterized by a nonlinear boundary condition. The nonlinearity, given by f(u)−auα with α>0, exhibits a singularity at u=0 and tends towards −∞ as u approaches 0+. By constructing some suitable auxiliary problems, the difficulty that arises from the singularity and semipositone of nonlinearity and the lack of a maximum principle is overcome. Subsequently, employing the Krasnosel’skii fixed-point theorem, we determine the parameter range that ensures the existence of at least one positive solution and the emergence of at least two positive solutions. Furthermore, based on our existence results, one can obtain the symmetry of the solutions after adding some symmetric conditions on the given functions by using a standard argument.

Bifurcation and dynamics of periodic solutions of MEMS model with squeeze film damping

In this paper, we study the oscillations of an idealized mass–spring model of micro-electro-mechanical system (MEMS) with squeeze film damping. The model consists of two parallel electrodes separated by a gap d: one of them is fixed, and another one is movable and attached to a linear spring with stiffness coefficient k>0. The oscillation, under the influence of AC–DC voltage V(t)=vdc+vaccos2πTt, is ruled by the following singular differential equation my′′+[A(d−y)3+Ad−y]y′+ky=θ0A2V2(t)(d−y)2.Here, y is the vertical displacement of the moving plate (y is always assumed to be less than d), m>0 is its mass, A>0 is the electrode area, and θ0>0 is the absolute dielectric constant of vacuum. Taking d as the parameter, we show the existence of saddle–node bifurcation of T-periodic solutions to the equation in the parameter space. This answers, from certain point of view, the open problem proposed by Torres in his monograph, see Torres (2015, Open Problem 2.1, p. 18). Further, we prove that the equation has exactly two classes of T-periodic solutions: as d tends to +∞, one of them uniformly tends to +∞ at the rate of d, while the minimum values of the second class tend to, or cross, 0.

A Two-Dimensional Nonlocal Fractional Parabolic Initial Boundary Value Problem

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments.

Design of integrated evolutionary finite differences for nonlinear electrohydrodynamics ion drag flow in cylindrical conduit model

This research implements an evolutionary optimized finite differences scheme (FDS) for nonlinear electrohydrodynamics ion drag flow dynamics in a cylindrical conduit (EHD-IDFCC). In the scheme, finite differences are exploited to discretize governing expressions of EHD-IDFCC, represented with a nonlinear singular seconder order differential equation, into a nonlinear equations-based system. The fitness function based on residual error is constructed for the FDS-based discretized EHD-IDFCC model. Its optimization is conducted with the global search capability of genetic algorithms (GAs) aided by the local search efficacy of the interior-point method (IPM), i.e., FDS-GA-IPM. The performance of the designed stochastic numerical solver FDS-GA-IPM is evaluated for a variant of the EHD-IDFCC model for different scenarios to measure the effect of axial flow velocity by varying the electric Hartmann numbers, as well as the nonlinearity factor. Statistical interpretations based on histogram illustrations, probability plots, and boxplots in terms of the cost function, mean absolute error (MAE), Thai inequality coefficients (TIC), and coefficient of determination metrics (R2) are used to endorse the accuracy, convergence, stability, and strength of the FDS-GA-IPM.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

A polynomial collocation method for a class of singular fractional differential equations

In this work we consider a class of singular fractional differential equations (SFDEs). Using a suitable variable transformation we rewrite the SFDE as a cordial Volterra integral equation and propose a polynomial collocation method to find an approximate solution of the original problem. We provide the error analysis of the numerical method and we illustrate its feasibility and accuracy through some numerical examples.

Mini-Workshop: Combinatorial and Algebraic Structures in Rough Analysis and Related Fields

Recent years have seen an explosion of algebraic methods to study singular stochastic and rough dynamics. These include developments in geometric rough path theory based on the algebra of words, the introduction of decorated trees in regularity structures, and the recent approach to singular stochastic partial differential equations based on multi-indices. These developments have furthermore led to important links with numerical analysis, machine learning, stochastic quantisation, and the study of symmetries of physical systems. The aim of this mini-workshop was to bring together experts working on these fields using algebraic structures that appear in rough dynamics. The goal was to facilitate the exchange of ideas and encourage further connections to be established.

Boundary value problems for integro-differential and singular higher-order differential equations

Abstract We investigate third-order strongly nonlinear differential equations of the type ( Φ ( k ( t ) u ″ ( t ) ) ) ′ = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t){u}^{^{\prime\prime} }\left(t)))^{\prime} =f\left(t,u\left(t),u^{\prime} \left(t),{u}^{^{\prime\prime} }\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], where Φ \Phi is a strictly increasing homeomorphism, and the non-negative function k k may vanish on a set of measure zero. Using the upper and lower solution method, we prove existence results for some boundary value problems associated with the aforementioned equation. Moreover, we also consider second-order integro-differential equations like ( Φ ( k ( t ) v ′ ( t ) ) ) ′ = f t , ∫ 0 t v ( s ) d s , v ( t ) , v ′ ( t ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t)v^{\prime} \left(t)))^{\prime} =f\left(t,\underset{0}{\overset{t}{\int }}v\left(s){\rm{d}}s,v\left(t),v^{\prime} \left(t)\right),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], for which we provide existence results for various types of boundary conditions, including periodic, Sturm-Liouville, and Neumann-type conditions.

Nonlinear Contractions Employing Digraphs and Comparison Functions with an Application to Singular Fractional Differential Equations

After the initiation of Jachymski’s contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem.

Bounded and homoclinic-like solutions of second-order singular difference equations

Bounded and homoclinic-like solutions of second-order singular difference equations

Spectral Solutions of Specific Singular Differential Equations Using A Unified Spectral Galerkin-Collocation Algorithm

This paper presents a novel numerical approach to addressing three types of high-order singular boundary value problems. We introduce and consider three modified Chebyshev polynomials (CPs) of the third kind as proposed basis functions for these problems. We develop new derivative operational matrices for the three modified CPs of the third kind by deriving formulas for their first derivatives. Our approach follows a unified method for numerically handling singular differential equations (DEs). To transform these equations into algebraic systems suitable for numerical treatment, we employ the collocation method in combination with the introduced operational matrices of derivatives of the modified CPs of the third kind. We address the convergence examination for the three expansions in a unified manner. We present numerous numerical examples to demonstrate the accuracy and efficiency of our unified numerical approach.

The Eigenvalue Properties of a Kind of Singular Differential Equations in 3-Dimensional Space

In this paper, we consider the eigenvalue problem of the singular differential equation [see formula in PDF] in a bounded open ball with Dirichlet boundary condition in 3-dimensional space, where, [see formula in PDF][see formula in PDF], [see formula in PDF] is a given constant[see formula in PDF]. And we have made a detailed characterization of the weak solution space. Furthermore, the existence of the minimum eigenvalue and the fundamental gap are provided.

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.

Bogoliubov-type recursions for renormalisation in regularity structures

Hairer's regularity structures transformed the theory of singular stochastic partial differential equations and their solutions. The notions of positive and negative renormalisation are central and the intricate interplay between these two renormalisation procedures is captured through the combination of cointeracting bialgebras and an algebraic Birkhoff-type decomposition of bialgebra morphisms. This work revisits the latter by defining Bogoliubov-type recursions similar to Connes and Kreimer's formulation of BPHZ renormalisation. We then apply our approach to the renormalisation problem for SPDEs.