Singular Equations Research Articles

Singular kinetic equations and applications

Singular kinetic equations and applications

Modified Integral Transform for Solving Benney-Luke and Singular Pseudo-Hyperbolic Equations

Abstract In this article, we propose a technique based on modified double integral transforms used to solve certain equations of materials science, namely Benney–Luke (BL) and singular pseudo-hyperbolic (SP-H) equations. We have established some analytical results. This method can provide accurate one-step solutions, although the equations used may exhibit a singularity in the initial conditions. Some numerical examples have been discussed for illustration and to show the effectiveness of the technique for certain types of equations. We have developed an exact solution in just one step, whereas other approaches require several stages to succeed in a particular solution, making the proposed strategy particularly successful and straightforward to apply to various varieties of the B–L and SP-H equations.

Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations

Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations

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.

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.

Existence of two infinite families of solutions for singular superlinear equations on exterior domains

In this article we study radial solutions of \(\Delta u + K(|x|) f(u) =0\) inthe exterior of the ball of radius \(R>0\) in \(\mathbb {R}^{N}\) with \(N>2\) where \(f\) grows superlinearly at infinity and is singular at \(0\) with \(f(u) \sim \frac{1}{|u|^{q-1}u}\) and \(0<q<1\) for small \(u\).We assume \(K(|x|) \sim |x|^{-\alpha}\) for large \(|x|\) and establish existence of two infinite families of sign-changing solutions when \(N+q(N-2) <\alpha <2(N-1)\). For more information see https://ejde.math.txstate.edu/Volumes/2024/06/abstr.html

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.

Blow-up of Solutions for a Problem with Balakrishnan-Taylor Damping and Nonlocal Singular Viscoelastic Equations

In this paper, we study the nonlinear one-dimensional viscoelastic nonlocal problem with Balakrishnan-Taylor damping terms and nonlinear source of polynomial type. We demonstrate that the nonlinear source of polynomial type is able to force solutions to blow up infinite time even in presence of stronger damping with non positive initial energy combined with a positive initial energy.

Existence of Positive Solutions for a Singular Hessian Equation with a Negative Augmented Term

In this paper, we focus on the existence of positive solutions for a singular Hessian equation with a negative augmented term. By finding more appropriate upper and lower solutions, we not only overcome the difficulty due to the negative augmented term but also remove a critical condition required in the existing work and establish new results for the existence of positive solutions of the equations under study. Our results improve and complement many existing works.

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

The numerical solution of fuzzy singular Lyapunov matrix equations

Fuzzy singular Lyapunov matrix equations have many applications, but feasible numerical methods to solve them are absent. In this paper, we propose an efficient numerical method for fuzzy singular Lyapunov matrix equations, where A is crisp and semi-stable. In our method, we transform fuzzy singular Lyapunov matrix equation into two crisp Lyapunov matrix equations. Then we solve the least squares solutions of the two crisp Lyapunov matrix equations, respectively. The existence of fuzzy solution is also considered. At last, two small examples are presented to illustrate the validate of the method and two large scale examples that the existing method fails to slove are presented to show the efficiency of the method.

Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations

<abstract><p>This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order $ \digamma $. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.</p></abstract>

SOLUTION TO SINGULAR HEAT EQUATION

Abstract. The main goal of this paper is to analyze the solution to the singular heat equation. A singular heat equation is the parabolic equation, where Laplace-Bessel operator acts by spatial variables

Method of modeling conformal radiating structures with chiral filling

At the moment, there is a scientific and technical problem of developing adequate methods and models of conformal antennas based on chiral metamaterials that allow for high accuracy of calculations with small computing resources. The aim of the work is to develop a method and mathematical models of conformal radiating structures based on the apparatus of singular equations that allow them to correctly calculate their characteristics. As a result, an analysis method and a mathematical model of conformal radiating structures with chiral filling were developed. A singular integral equation is obtained with respect to an unknown function of the longitudinal distribution of the current density. The advantage of the developed method is the possibility of correct calculation of the characteristics of conformal radiating structures with filling from chiral metamaterials. The results obtained can be used to create conformal antennas based on chiral metamaterials.

Analyzing the structure of solutions for weakly singular integro-differential equations with partial derivatives

<p>In this work, we analyze the approximate solution of a specific partial integro-differential equation (PIDE) with a weakly singular kernel using the spectral Tau method. It present a numerical solution procedure for this PIDE, which is transferred into a Volterra–Fredholm integral equation (VFIE), and the spectral method is performed on VFIE. In some illustrated examples, we show that the VFIE problem has high numerical stability with respect to the original form of the PIDE problem. For this aim, we apply the spectral Tau method in two cases, first for the problem in the form of VFIE and then also for the problem in the form of PIDE. The remarkable numerical results obtained from the VFIE problem form compared to those gained from the PIDE problem form show the efficiency of the proposal method. Also, we prove the convergence theorem of the numerical solution of the Tau method for the VFIE problem, and then it is generalized to the PIDE problem.</p>

Multiple positive solutions for a singular tempered fractional equation with lower order tempered fractional derivative

<abstract><p>Let $ \alpha\in (1, 2], \beta\in (0, 1) $ with $ \alpha-\beta > 1 $. This paper focused on the multiplicity of positive solutions for a singular tempered fractional boundary value problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{aligned}\ & -{^R _0}{{\mathbb{D}_t}^{\alpha,\lambda}} u(t) = p(t)h\left(e^{\lambda t} u(t), {^R _0}{{\mathbb{D}_t}^ {\beta,\lambda}}u(t)\right), t\in(0,1),\\& {^R _0}{{\mathbb{D}_t}^ {\beta,\lambda}}u(0) = 0, \ \ {^R _0}{{\mathbb{D}_t}^ {\beta,\lambda}}u(1) = 0, \end{aligned}\right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ h\in C([0, +\infty)\times[0, +\infty), [0, +\infty)) $ and $ p \in L^1([0, 1], (0, +\infty)) $. By applying reducing order technique and fixed point theorem, some new results of existence of the multiple positive solutions for the above equation were established. The interesting points were that the nonlinearity contained the lower order tempered fractional derivative and that the weight function can have infinite many singular points in $ [0, 1] $.</p></abstract>

Resistive instabilities in general toroidal plasmas with neoclassical bootstrap currents

In this work, linear neoclassical resistive instabilities are investigated in general toroidal plasmas using standard perturbation theory. Using a neoclassical fluid model, we derive the singular layer equations modified by bootstrap currents and also obtain the dispersion relation of the resistive interchange mode and the neoclassical tearing mode (NTM), respectively. Additionally, we determine the stability criteria DRbs and Δcbs for bootstrap current-modified resistive modes. The resistive interchange mode is stable when DRbs<0 and the NTM is stable when Δ′<Δcbs, where Δ′ is the stability index of the tearing mode. It is found that, in tokamak plasmas with a positive magnetic shear, bootstrap currents have a destabilizing effect on resistive interchange modes, which not only increases the value of the stability criterion (DRbs) but also enhances the growth rate. In addition, bootstrap currents have a stabilizing effect on the growth rate of the NTM in a low growth rate region. However, bootstrap currents can also decrease the critical value Δcbs. In plasmas with negative magnetic shear, the opposite holds. Furthermore, the coupling effect between bootstrap currents and Pfirsch–Schlüter currents via magnetic field curvature is determined for the first time in this work. This coupling always has a stabilizing influence on the resistive interchange mode and can increase the value of Δcbs. The coupling is also independent of the sign of the magnetic shear and can be enhanced in low-aspect-ratio tokamaks (such as spherical tokamaks) or in plasma regions with low magnetic shear (as used in ITER hybrid scenarios). Our results are valid for low-n resistive instabilities in toroidal plasmas with arbitrary aspect ratios and β, where n is the toroidal mode number and β represents the ratio of the plasma pressure to the toroidal magnetic pressure. Overall, this investigation has broad parametric applications and deepens our understanding of the physical mechanisms underlying the influence of neoclassical effects on resistive instabilities.

Double‐Scale Expansions for a Logarithmic Type Solution to a q‐Analog of a Singular Initial Value Problem

We examine a linear q−difference differential equation, which is singular in complex time t at the origin. Its coefficients are polynomial in time and bounded holomorphic on horizontal strips in one complex space variable. The equation under study represents a q−analog of a singular partial differential equation, recently investigated by the author, which comprises Fuchsian operators and entails a forcing term that combines polynomial and logarithmic type functions in time. A sectorial holomorphic solution to the equation is constructed as a double complete Laplace transform in both time t and its complex logarithm logt and Fourier inverse integral in space. For a particular choice of the forcing term, this solution turns out to solve some specific nonlinear q−difference differential equation with polynomial coefficients in some positive rational power of t. Asymptotic expansions of the solution relatively to time t are investigated. A Gevrey‐type expansion is exhibited in a logarithmic scale. Furthermore, a formal asymptotic expansion in power scale is displayed, revealing a new fine structure involving remainders with both Gevrey and q−Gevrey type growth.

APPLICATION OF THE POTENTIAL THEORY TO SOLVING A MIXEDPROBLEM FOR A MULTIDIMENSIONAL ELLIPTIC EQUATION WITHONE SINGULAR COEFFICIENT

Potential theory has played a paramount role in both analysis and computation for boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using some properties of Gaussian hypergeometric function, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find a solution of the mixed problem for the multidimensional singular elliptic equation in the half-space