Nonhomogeneous Equations Research Articles

On a localization-in-frequency approach for a class of elliptic problems with singular boundary data

We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

Effect of an edge dislocation on hydride growth in zirconium: elastic complex potential study

The formation of hydrides can significantly decrease the mechanical reliability of Zr servicing in the nuclear reactor. This study investigates the effect of the local stress field resulting from the hydride eigenstrain and dislocations on hydrogen transport and explores the hydride growth regulation. The stress functions, which can be extended and further applied to the realistic crystal structure and servicing status, are derived using the dislocation-based strain nucleus model and elastic complex potential method. The nonhomogeneous equation of hydrogen transport governed by diffusion and hydrostatic stress gradient is analytically solved. The results show that: (1) In the absence of dislocations, the hydride grows toward 112¯0 direction under the hydrostatic stress gradient resulting from the hydride eigenstrain. (2) The lower the temperature is, the higher the hydrogen concentration gradient in the adjacent area of the hydride is. The higher hydrogen fraction can also accelerate the growth rate. (3) Dislocation defects are capable of inducing triangle-shaped and asymmetric growth tendencies and providing new nucleation loci for hydrides. Based on the significant hydrostatic stress gradient led by dislocation singularity, this study analytically explains the autocatalysis and new-nucleation-locus effects raised by the in-situ observations.

Research on finite element model of rotor aerodynamicnoise in hovering state

The helicopter has been widely used in many fields, and the rotor aerodynamic noise has gradually attracted attention. The non-homogeneous wave equation, Ffowcs Williams-Hawkings equation (FW-H equation) derived from hydrodynamic N-S equation, combined with RANS turbulence simulation, was used to simulate the sound field of NACA0012 airfoil-shape model in hovering state. On this basis, by changing the thickness and camber of the blade, the aerodynamic noise of the rotor under different airfoil profile thickness and camber is calculated. The results show that the change of airfoil thickness and camber will affect the aerodynamic noise of the rotor in hovering state.

Existence conditions for 2-periodic solutions to a non-homogeneous differential equations with piecewise constant argument

This paper provides a method of finding 2-periodical solutions for the first-order non-homogeneous differential equations with piecewise constant arguments. All existence conditions are described for 2-periodical solutions and obtained explicit formula for these solutions. An example for the problem that has infinitely many solutions is constructed.

A Novel Meshfree Method for Nonlinear Equations in Flow through Porous Media and Electrohydrodynamic Flows

In this study, an efficient meshfree numerical method is introduced for solving the nonlinear boundary value problems. The method of fundamental solutions (MFS) is one of the most popular among meshfree methods. While traditionally limited to linear and homogeneous problems, this study extends the applicability of the MFS to include nonhomogeneous and nonlinear equations. To achieve this, an extended MFS is combined with a fixed-point iteration scheme. This developed framework is benchmarked to address two different flow problems. The first involves fluid flow through porous media in a channel governed by the nonlinear Brinkman-Forchheimer equation. The second problem pertains to electrohydrodynamic (EHD) flow in a circular conduit. The obtained solutions are compared with the finite element method and the solutions available in the existing literature.

Solution to Fourier Transforms Method to Complex Variable of Non-homogeneous Fractional Differential Equations

This work is devoted to the study of fractional differential equations involving Caputo non-homogeneous fractional differential equations. Using Fourier transform method, a complex variable explicit solution to non-homogeneous second-order fractional differential equation was obtained.

Hölder continuity of the gradients for non-homogenous elliptic equations of p(x)-Laplacian type

Hölder continuity of the gradients for non-homogenous elliptic equations of p(x)-Laplacian type

Theoretical estimation of the thermal damages of living tissues caused by laser irradiation in tumor thermal therapy

This article aims to provide theoretical predictions for the thermal reactions of human tissues during tumor thermotherapy when exposed to laser irradiation and an external heat source. For the construction of a theoretical study of bioheat transfer, the selection of a suitable thermal model capable of accurately predicting the required thermal responses is essential. The effect of heat production by heat treatment on a spherical multilayer tumor tissue is evaluated using this approach. Analytical solution for the non-homogenous differential equations is derived in the Laplace domain. The study examines the impact of thermal relaxation time on tissue temperature and the subsequent thermal damage. The numerical findings of thermal damage and temperatures are depicted in a graphical representation. This model explains laser treatment, physical events, metabolic support, and blood perfusion. The numerical outcomes of the recommended model are validated by comparing them to the literatures.

Tackling Ternary Cubic Diophantine Equation

Abstract: A few interesting characteristics among the solutions and the patterns of non-zero integral solutions to the nonhomogeneous cubic equation with three unknowns represented by the Diophantine equation 2 2 3 4(   )  7( )  64 are examined.

Holistic Solutions of Cubic Diophantine Equation with Four unknowns

The non-homogenous cubic Diophantine equation with four unknowns represented

METODA OBLICZANIA WSKAŹNIKA BEZPIECZEŃSTWA INFORMACJI W MEDIACH SPOŁECZNOŚCIOWYCH Z UWZGLĘDNIENIEM DŁUGOŚCI ŚCIEŻKI MIĘDZY KLIENTAMI

Heterogeneous differential dependencies of the information security indicator (HDISI) in social media (SM) were analyzed, taking into account the duration of the path between clients (UDPC). The resilience of the information security indicator system (RSIIS) was also determined. The HDISI in SM was developed based on UDPC conditions. It uses modern methods and techniques, including a non-specific method. The conditions of a fixed precondition were formed according to the time grid. This dependency provides a comprehensive explanation of how the previous transformation is replaced by the elapsed period. SM is a set of clients and their types of communication. Clients can be individuals, populations, settlements, or countries. Communication is understood as more than just the transmission and receipt of information. It also includes interaction, the exchange of knowledge and expertise, and discussion. Under the angle of mathematics, the HDISI model based on non-homogeneous differential equations (NDE) was analyzed and its transcendental study was done. The transcendental study of nonlinear HDISI models in SM showed that the characteristics of UDPC significantly affect the information security indicator (ISI) - up to one hundred percent. Phase diagrams (PDs) of ISI were studied, which indicate the highest ISI even at the maximum parameters of malicious actions. For the first time, the analysis of designed HDISI structures was carried out and numerical criteria between the capabilities of UDPC and the measures of ISI, as well as the highest ISI, were obtained, which shows the scientific content of this article.

Analyze a temperature and MHD peristaltic flow of sutterby fluid through a porous wavechannel in a rotating frame

The aim of this paper is to analyze the temperature and magnetohydrodynamic (MHD) peristaltic flow of non-Newtonian Sutterby fluid through a porous wavechannel under the influence of rotation. The problem formula is non-linear and non-homogeneous partial differential equations. By using the perturbation technique, we solved the momentum equation under the assumptions of a long wavelength and an extremely low Reynolds number. After getting the solution, we have used the "Mathematica 13" program to analyze the results through graphs, and it have been studied the effect of the rotation, magnetic field, viscosity of the fluid and average radius of the tube on the fluid movement and its temperature.

A Solution of Fractal Dynamic Pharmacokinetics Problem

There are numerous uses for fractional differential equations in engineering, physics, and technology. A path for solving the fractal differential equations was examined, and its homogeneous form of was introduced. The notions of Riemann-Liouville fractional derivatives served as the foundation for the journey.. The solutions of the linear non-homogeneous fractal differential equations are given in detail.

A homotopy-based computational scheme for two-dimensional fractional cable equation

In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach’s fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering.

Applying Proper Orthogonal Decomposition to Parabolic Equations: A Reduced Order Numerical Approach

n this paper we present a low-order numerical scheme developed using the Proper Orthogonal (POD) method to address non-homogeneous parabolic equations in both one and two dimensions. The proposed schemes leverage the POD technique to reduce the computational complexity associated with solving these equations while maintaining accuracy. By employing POD, the high-dimensional problem is approximated by a reduced set of models, allowing for a more efficient representation of the system dynamics. The application of this method to non-homogenous parabolic equations offers a promising approach for enhancing the computational efficiency of simulations in diverse fields, such as fluid dynamics, heat conduction, and reaction-diffusion processes. The presented numerical scheme demonstrates its efficacy in achieving accurate results with significantly reduced computational costs, making it a valuable tool for applications demanding efficient solutions to non-homogeneous parabolic equations in one and two dimensions.

Cobalancing Numbers: Another Way of Demonstrating Their Properties

In this study, previously obtained cobalancing numbers are considered from a different perspective, and the properties of the numbers are re-examined. The main purpose is to change the recurrence relation of cobalancing numbers and calculate some relations and properties in a more diverse and easier way. The reason that led us to this method is that the recurrence relation of cobalancing numbers has a second-order but non-homogeneous difference equations. Thus, it will be much easier to find the Binet formula, generating function, sum formulas, and many other relations with a sequence that is homogeneous and has a third-degree recurrence relation. Also some identities that have not been found before in the sequence are also included in this study.

The concentration and cavitation in the Riemann solution for non-homogeneous logrithmic equation of state with magnetic field

In the present study, the phenomena of concentration and cavitation in the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field is analyzed. Firstly, we introduced new state variable for the velocity to modify the non-conservative system into conservative system and solved the Riemann problem for modified system constructively. Further, the Riemann solutions for the transport equations is investigated as pressure and magnetic field vanish. It is proved that the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field having two shock waves converges to the delta shock wave solution of the transport equations as pressure and magnetic field vanish. It is also proved that the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field having two rarefaction waves converges to the contact discontinuity solution of the transport equations as pressure and magnetic field vanish.

Inverse Problem for a Non-Homogeneous Integro-Differential Equation of the Hyperbolic Type

Inverse Problem for a Non-Homogeneous Integro-Differential Equation of the Hyperbolic Type

ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NON-HOMOGENEOUS SECOND-ORDER DIFFERENCE EQUATION OF ACCRETIVE TYPE IN 2-BANACH SPACES

In this study we present the existence and uniqueness of a non-homogeneous second order difference equation of accretive type in linear 2-Banach spaces. We use the 2-norm contraction mapping and its fixed point to establish our results.

Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space

The paper is concerned with positive solutions to problems of the type \[ -\Delta_{\mathbb{B}^{N}} u - \lambda u = a(x) |u|^{p-1}\;u + f \text{ in }\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \] where $\mathbb {B}^N$ denotes the hyperbolic space, $1< p<2^*-1:=\frac {N+2}{N-2}$ , $\;\lambda < \frac {(N-1)^2}{4}$ , and $f \in H^{-1}(\mathbb {B}^{N})$ ( $f \not \equiv 0$ ) is a non-negative functional. The potential $a\in L^\infty (\mathbb {B}^N)$ is assumed to be strictly positive, such that $\lim _{d(x, 0) \rightarrow \infty } a(x) \rightarrow 1,$ where $d(x,\, 0)$ denotes the geodesic distance. First, the existence of three positive solutions is proved under the assumption that $a(x) \leq 1$ . Then the case $a(x) \geq 1$ is considered, and the existence of two positive solutions is proved. In both cases, it is assumed that $\mu ( \{ x : a(x) \neq 1\}) > 0.$ Subsequently, we establish the existence of two positive solutions for $a(x) \equiv 1$ and prove asymptotic estimates for positive solutions using barrier-type arguments. The proofs for existence combine variational arguments, key energy estimates involving hyperbolic bubbles.