Singular Equations Research Articles

Analytical and Numerical Methods for the Solution to the Rigid Punch Contact Integral Equations

In this article, the problem of a rigid punch pressed onto the surface of an elastic half-plane is studied. In a first instance, it is reminded that the standard frictionless hard contact situation, where the contact pressure is unbounded at contact ends, exhibits an analytical solution to the governing singular integral equation with Cauchy kernel. Thereafter, it is shown that the situation of contact regularization results in a singular integro-differential equation with Cauchy kernel. This latter case leads to bounded contact pressures at both contact ends, even in the frame of linear elasticity, which is of great interest in the presence of “peaking” phenomenon. However, this contact regularization requires a numerical treatment as opposed to the former. To that end, a novel simple but efficient numerical procedure, based on numerical integration in conjunction with a centered finite differences scheme, is presented and numerically illustrated through two examples at the end of the paper.

Gauss quadrature rules for integrals involving weight functions with variable exponents and an application to weakly singular Volterra integral equations

Abstract This paper presents a numerical integration approach that can be used to approximate on a finite interval, the integrals of functions that contain Jacobi weights with variable exponents. A modification of the integrand close to the singularities is needed, and a new modification is proposed. An application of such a rule to the numerical solution of variable-exponent weakly singular Volterra integral equations of the second kind is also explored. In the space of continuous functions, the stability and the error estimates are demonstrated, and numerical tests that validate these estimates are conducted.

Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with anomalous scaling

Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with anomalous scaling

Exact controllability for degenerate and singular wave equations

Exact controllability for degenerate and singular wave equations

Singular fractional integro-differential equations with non-constant coefficients

Singular fractional integro-differential equations with non-constant coefficients

Second-order maximum principle controlled weakly singular Volterra integral equations

This work studies a class of singular Volterra integral equations that are (controlled) and can be applied to memory-related problems. For optimum controls, we prove a second-order Pontryagin type maximal principle.

Multi‐indice B$B$‐series

AbstractWe propose a novel way to study numerical methods for ordinary differential equations in one dimension via the notion of multi‐indice. The main idea is to replace rooted trees in Butcher's ‐series by multi‐indices. The latter were introduced recently in the context of describing solutions of singular stochastic partial differential equations. The combinatorial shift away from rooted trees allows for a compressed description of numerical schemes. Furthermore, such multi‐indices ‐series uniquely characterize the Taylor expansion of 1‐dimensional local and affine equivariant maps.

INITIAL PROBLEMS FOR THE ABSTRACT LEGENDRE EQUATION CONTAINING TWO PARAMETERS

Using the concept of a fractional integral of a function over another function, transformation operators are constructed that make it possible to prove the solvability of initial problems for the abstract singular Legendre equation containing two parameters. Examples are given.

Generalized solutions for singular double-phase elliptic equations under mixed boundary conditions

In this article, we investigate at least one or two generalized solutions for double-phase singular elliptic equations with Hardy potential. We show the existence of at least one or two distinct generalized solutions under mixed boundary conditions via variational methods when the nonlinearity f satisfying suitable hypotheses.

Dynamic analysis of the air–water interface instability in a wind-wave flume: Initial conditions for wave generation

This paper aims to investigate the dynamic mechanism for a previously calm water surface under a background wind field, which would lose stability due to minor perturbations. When spatial distribution and growth rate of the shear wind field satisfy a certain relationship, an Orr–Sommerfeld equation can be derived from linearized Navier–Stokes equations, for the amplitude of perturbation waves at the air–water interface. After transforming its coordinates to a finite interval, this fourth-order linear ordinary differential equation with variable coefficients can turn into a linear singular differential equation. It is solved under the background flow fields of air and water using the corrected Fourier method that we previously developed, to yield two sets of four analytical solutions with physical significance. The perturbation flow at the air–water interface must satisfy the kinematic and dynamic matching conditions (continuity of velocity and smooth transition of shear stress), which results in a fourth-order homogeneous linear algebraic equation for four undetermined constants. Setting various parameters in accordance with measured data with a wind-wave flume, the corresponding perturbation waves solution is identified. Our calculation densely distributes the most unstable and readily growing waves among perturbation waves with small phase speeds, consistent with the experimental results. Our perturbation solutions over a flat air–water interface are only applicable to the short period after instability onset, which provides an initial condition for wave generation. The consequent interaction between small-amplitude waves and wind field is beyond the scope of our linear theory, which has indeed been sufficiently addressed in the literature.

Singular double phase equations with a sign changing reaction

Singular double phase equations with a sign changing reaction

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.

Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients

Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients

A Matrix-Based Approach to Unified Synthesis of Planar Four-Bar Mechanisms for Motion Generation With Position, Velocity, and Acceleration Constraints

Abstract This paper introduces a novel matrix-based approach for the simultaneous type and dimensional synthesis of planar four-bar linkage mechanisms, accommodating various practical constraints, including position, velocity, acceleration, and joint placements. Traditional design processes segregate type synthesis, the determination of joint and link configurations, from dimensional synthesis, which involves specifying link sizes and pivot locations. This segregation often leads to complexities in addressing the complete design challenge. The novel methodology proposed in this paper departs from the conventional sequential design approach by concurrently evaluating type and dimensional parameters using a data-driven matrix formulation. The crux of the paper’s methodology involves formulating a singular design equation through a transformation matrix, parameterized by the Cartesian parameters of the mechanism’s dyads. This formulation linearly expresses a broad range of constraints, facilitating the identification of viable solutions through singular value decomposition and null space analysis. This integrated approach not only simplifies the synthesis process but also provides direct insights into the mechanism’s parameters, encompassing both type and dimensions, thereby obviating the need for further interpretative steps common to the use of quaternions and kinematic mapping. In essence, the paper presents two main contributions: the development of a unified design equation capable of encompassing a wide array of constraints within the mechanism synthesis process, and the introduction of an algorithm that effectively identifies all potential planar four-bar linkage mechanisms by accurately satisfying up to five constraints. This approach promises to enhance the design and optimization of mechanical systems by offering a more holistic and efficient pathway to mechanism synthesis.

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.

The Uniqueness and Iterative Properties of Positive Solution for a Coupled Singular Tempered Fractional System with Different Characteristics

In this paper, we focus on the uniqueness and iterative properties of positive solution for a coupled p-Laplacian system of singular tempered fractional equations with differential order and characteristics. Firstly, the system is converted to an integral equation, and then, a coupled iterative technique and some suitable growth conditions are proposed; furthermore, some elaborate results about the uniqueness and iterative properties of positive solutions of the system are established, which include the uniqueness, the convergence analysis, the asymptotic behavior, and error estimation, as well as the convergence rate of the positive solution. The interesting points of this paper are that the order of the system of equations is different and the nonlinear terms of the system possess the opposite monotonicity and allow for stronger singularities at space variables.

Simultaneous identification of the initial data in a degenerate and singular hyperbolic wave equation

Simultaneous identification of the initial data in a degenerate and singular hyperbolic wave equation

An efficient temporal approximation for weakly singular time-fractional semilinear diffusion-wave equation with variable coefficients

An efficient temporal approximation for weakly singular time-fractional semilinear diffusion-wave equation with variable coefficients

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.