Singular Coefficients Research Articles

Development of mathematical models of heat conductivity for modern electronic devices with elements containing foreign inclusions

This paper considers the heat conduction process for an isotropic medium containing a foreign half-through inclusion and heated by a locally concentrated heat flow. Linear and non-linear mathematical models for determining the temperature field have been built to establish the temperature regimes for the effective operation of electronic devices. The coefficient of thermal conductivity of a non-uniform structure is represented as a whole, using asymmetric unit functions, which automatically provides the conditions of ideal thermal contact on the surfaces of materials. This results in solving one heat conduction equation with discontinuous and singular coefficients. A linearizing function was introduced to linearize the nonlinear boundary value problem. Analytical-numerical solutions of linear and nonlinear boundary-value problems have been obtained in a closed form. A linear temperature dependence of the coefficient of thermal conductivity of structural materials was chosen for a heat-sensitive medium. As a result, an analytical-numerical solution was derived, which determines the temperature distribution in this medium. On this basis, a numerical experiment was performed, the results of which are graphically displayed and confirm the adequacy of the constructed mathematical models to a real physical process. The materials of the plate and inclusion are silicon and silver. The results for these materials based on the linear and non-linear model differ by 7 %. Their slight difference is explained by the fact that the values of the temperature coefficient of thermal conductivity are small. The models built make it possible to analyze the given environments in terms of their thermal resistance. As a result, it becomes possible to improve it, and protect structures from overheating, which could lead to the failure of individual nodes and their elements and the entire electronic device

Maximum principle for non-uniformly parabolic equations and applications

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion and drift coefficients. Moreover, by the Markov selection theorem of Krylov [8], we also establish the existence of the associated strong Markov family.

Heat equation for Sturm–Liouville operator with singular propagation and potential

Abstract This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm–Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm–Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

Evolution of coupled scalar perturbations through smooth reheating. Part I. Dissipative regime

If the inflaton is a heavy scalar field, it may equilibrate slower thansome other degrees of freedom, e.g. non-Abelian gauge bosons. In this case,perturbations in the inflaton field and in a thermalplasma coexist from a given moment onwards. We derive a gauge-invariant setof three coupled equations governing the time evolution of such a system.Despite singular coefficients,a reliable numerical solution can be obtained for a longtime period, starting from phase oscillations inside the Hubble horizon,and extending until acoustic oscillations in a radiation-dominated universe.Benchmarks are illustrated from a “weak regime”,where perturbations have a quantum-mechanical origin but get dissipatedby interactions with the plasma. Among applications of our formalismcould be inhomogeneity-induced nucleations inpost-inflationary phase transitions, andthe production of scalar-induced gravitational waves.

Analysis of degenerate [formula omitted]-Laplacian elliptic equations involving Hardy terms: Existence and numbers of solutions

This article investigates the existence of solutions to quasilinear degenerate elliptic equation with Hardy singular coefficient, in which the weighted function ω(x) is unbounded (singular), then we cannot use the classical space W01,p(Ω), so we have to find another space W01,p(ω,Ω) to deal with the difficulties caused by singularities or degeneracies. New criteria for the existence of at least one and at least two generalized solutions are established via variational methods and critical point theorems provided that the nonlinearity satisfies appropriate hypotheses.

P(x)-biharmonic equations involving (h,r(x))-Hardy singular coefficients with no-flux boundary conditions

In this article, we investigate $p(x)$-biharmonic equations involving two kinds of different Hardy potentials, in which the $r(x)$-Hardy potentials are seldom mentioned in previous papers. New criteria for the existence of generalized solutions are reestablished when the nonlinear terms satisfying appropriate assumptions. The results are based on variational methods and the theory of variable exponent Sobolev spaces.

Automatic classification method for water entry sounds in multi-interference environments

Abstract In marine environments with strong pulse interference, distinguishing water entry sounds from pulse interference with high accuracy presents a challenge. Conventional algorithms that only utilize the singular values of intrinsic mode functions (IMFs) as the classification feature can have many misclassified signals in noisy environments. To identify the water entry sound more precisely, this study introduces a classifier based on selected singular value and correlation coefficient features. Specifically, the correlation coefficient between each IMF and the original signal are used to select crucial singular values and eliminate uninformative components. Furthermore, the correlation coefficient also combines with singular values as classification features for water entry sounds. The classification experiment results of 2460 groups of water entry sound and pulse interference indicate that the proposed classifier improves classification accuracy by approximately 11.5% compared to using singular values alone and by approximately 2.4% compared to classification accuracy without eliminating uninformative IMFs.

Non-Local Problems for the Fractional Order Diffusion Equation and the Degenerate Hyperbolic Equation

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front.

Flexural Rigidity Influence on Dynamic Response of Orthoropic Rectangular Plate Resting on Constant Elastic Foundation

This research article considers the flexural rigidity influence along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions. The orthotropic rectangular plate model is a coupled fourth order partial differential equation having variables and singular coefficients. The solutions to this model are arrived at by reconstructing the fourth order partial differential. This partial differential equation model is converted to a set of coupled second order ordinary differential equations by using a special technique adopted by Shadnam et al., [11]. This set of second order ordinary differential equations is then reduced using modified asymptotic method of Struble. The closed form solution is evaluated, resonance conditions are obtained and the results are showed in plotted curves to depict the influence of flexural rigidities along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions for both cases of moving distributed mass and moving distributed force.

Amplitude Variations of Moving Distributed Masses of Orthotropic Rectangular Plate with Elastically Supported Ends under Moving Loads

The deflection of thin orthotropic rectangular plate under moving loads is a classic problem in solid mechanics. However, the equations are challenging to solve due to their non linearity and complexity. At the same time, this equation is a coupled fourth order partial differential equation having variables and singular coefficients. In this research article, the partial differential equation is converted to a set of coupled second order ordinary differential equations by using a special technique adopted by Shadnam et al., [19]. This transformed set of second order ordinary differential equations is then reduced using modified asymptotic method of Struble and Laplce transformation. The closed form solution is evaluated, resonance conditions are obtained and the results are showed in plotted curves to solve the variations in amplitudes for some varying orthotropic plate parameters with elastically supported ends under moving loads for both cases of moving distributed force and moving distributed mass.

Construction of mathematical models of thermal conductivity for modern electronic devices with elements of a layered structure

This paper considers a heat conduction process for isotropic layered media with internal thermal heating. As a result of the heterogeneity of environments, significant temperature gradients arise as a result of the thermal load. In order to establish the temperature regimes for the effective operation of electronic devices, linear and non-linear mathematical models for determining the temperature field have been constructed, which would make it possible to further analyze the temperature regimes in these heat-active environments. The coefficient of thermal conductivity for the above structures was represented as a whole using asymmetric unit functions. As a result, the conditions of ideal thermal contact were automatically fulfilled on the surfaces of the conjugation of the layers. This leads to solving one heat conduction equation with discontinuous and singular coefficients and boundary conditions at the boundary surfaces of the medium. For linearization of nonlinear boundary value problems, linearizing functions were introduced. Analytical solutions to both linear and nonlinear boundary value problems were derived in a closed form. For heat-sensitive environments, as an example, the linear dependence of the coefficient of thermal conductivity of structural materials on temperature, which is often observed when solving many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in these environments were obtained. Based on this, a numerical experiment was performed, and it was geometrically represented depending on the spatial coordinates. This proves that the constructed linear and nonlinear mathematical models testify to their adequacy to the real physical process. They make it possible to analyze heat-active media regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements but also the entire structure

Existence of solutions for singular elliptic equations with mixed boundary conditions

In this article, we investigate generalized solutions for elliptic equations involving double-phase ( p ( x ) , q ( x ) ) -Laplacian operators with Hardy potential in variable exponent spaces. ( p ( x ) , q ( x ) ) -Laplacian operators include p-Laplacian, q-Laplacian, p ( x ) -Laplacian and q ( x ) -Laplacian operators. Additionally, ( p ( x ) , q ( x ) ) -Laplacian elliptic equations with singular coefficients under mixed boundary conditions are seldom mentioned in previous work. Some new theorems of the existence on the generalized solutions are reestablished for such equations via variational methods when the nonlinearity satisfies suitable hypotheses in variable exponent Lebesgue spaces.

A Nonlocal Problem for a Degenerate Equation of Elliptic Type with a Singular Coefficient

A Nonlocal Problem for a Degenerate Equation of Elliptic Type with a Singular Coefficient

The problem with the missing Goursat condition at the boundary of the domain for a degenerate hyperbolic equation with a singular coefficient

The problem with the missing Goursat condition at the boundary of the domain for a degenerate hyperbolic equation with a singular coefficient

Lyapunov‐type inequality to general second‐order elliptic equations

We establish Lyapunov‐type inequality for equations concerning general class of second‐order non‐symmetric elliptic operators with singular coefficients. Our approach is based on the probabilistic representation of solutions and stochastic calculus. We also discuss a Lyapunov‐type inequality for equations pertaining to second‐order symmetric operator with some regularity assumptions on the coefficients and a nonlinear Neumann boundary condition.

A problem with analogue of the Frankl and mixing conditions for the Gellerstedt equation with singular coefficient

For the equation (sign y)| y| muxx + uyy + α0 | y| (m - 2)/2ux + (β0/y)uy = 0, $, considered in some unbounded mixed domain, uniqueness and existence theorems for a solution to the problem with the missing shift condition on the boundary characteristics and an analogue of the Frankl type condition on the interval of degeneracy of the equation are proved.

Quaternion Differential Matrix Equations with Singular Coefficient Matrices

Quaternion Differential Matrix Equations with Singular Coefficient Matrices

Exact solutions for some classes of non-elliptic Cauchy–Riemann type systems with singular coefficients

This paper deals with the solutions for some classes of non-elliptic Cauchy–Riemann type systems with singular coefficients.We give an explicit representation of the solutions and study their singularities

A Simple Approach to a Dirichlet Problem Related to the Kardar-Parisi-Zhang Equation with Singular Coefficients

A Simple Approach to a Dirichlet Problem Related to the Kardar-Parisi-Zhang Equation with Singular Coefficients

The Tricomi–Neumann Problem for a Three-Dimensional Mixed-Type Equation with Singular Coefficients

The Tricomi–Neumann Problem for a Three-Dimensional Mixed-Type Equation with Singular Coefficients