Nonhomogeneous Equations Research Articles

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.

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.

Simulation of exchange processes in multi-component environments with account of data uncertainty

The goal of the work. Proposals for methods of solving systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients that defined in interval form and intended for modeling exchange processes in multicomponent environments. Research subject: systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients defined in interval form. Research method: interval analysis. The obtained results. Systems of linear homogeneous and non-homogeneous differential equations, which are used in modeling exchange processes in multicomponent environments, are considered. Such systems can be considered, for example, in problems of chemical kinetics, materials science, and the theory of Markov processes. To obtain the solution of these equations, specialized calculators of analytical transformations were used and tested. The Matlab system (ode15s solver) was used for numerical analysis of systems of differential equations. It is shown that the application of interval methods of numerical analysis at the initial stage of system modeling has some advantages over probabilistic methods because they do not require knowledge of the laws of distribution of the results of the system state parameter measurements and their errors. It is shown that existing methods of solving systems of linear differential equations can be divided into two groups. Common to these groups is the use of interval expansion of classical methods for solving differential equations given in interval form. The difference between these two groups of methods is as follows. The methods of the first group can be used for all types of differential equations but require the creation of special software. The peculiarity of the methods of the second group is that they can be used to solve equations analytically or using numerical analysis packages. The application of the methods of the second group is shown on the example of solving a system of differential equations, the coefficients of which are determined in interval form. The system of these equations is intended for modeling the processes of exchange with the external environment of the elements of the model of a specific physical system. In the case when the coefficients of these equations are variables, their piecewise-constant approximation is applied and a criterion that determines the possibility of its application is given. The technique proposed in the paper can be applied to solve systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients if they are given by slowly varying functions. In the case when the coefficients of the equations are determined in the interval form, the technique allows obtaining their solution also in the interval form and does not require the creation of special software.

Solving Saint Venant torsion problems for rectangular beams using single finite Fourier sine transform method

This research presents the single Fourier sine transform method (SFSTM) for solving the Saint Venant torsion problem of rectangular prismatic bars. The problem is a common theme in the theory of elasticity of unrestrained torsion which was previously expressed by Prandtl using Prandtl stress functions ϕ(x,y) as a Poisson type nonhomogeneous partial differential equation (PDE) called the stress compatibility equation. In this work the SFSTM was applied to the stress compatibility equation, converting the PDE to an easier to solve ordinary differential equation (ODE) in the transformed domain. The boundary conditions were used to find the integration constant and inversion was used to find the solution in the physical domain. The non vanishing stresses and torsional moments were thus found as a single series of infinite terms with rapid convergence. The maximum stresses and moments were found in standard form in terms of torsional parameters which were tabulated for various ratios of the cross-sectional dimensions. A comparison of the torsional parameters with previous results show that the present results are identical with previous results illustrating the accuracy of the SFSTM used. The sine kernel of the SFSTM satisfies the boundary conditions of the problem and contributed to the exact solution obtained. The SFSTM simplified the PDE to an ODE which is simpler to solve.

Teukolsky-like equations in a non-vacuum axisymmetric type D spacetime

We study an axisymmetric metric satisfying the Petrov type D property with some additional ansatze, but without assuming the vacuum condition. We find that our metric in turn becomes conformal to the Kerr metric deformed by one function of the radial coordinate. We then study the gravitational-wave equations on this background metric in the case that the conformal factor is unity. We find that under an appropriate gauge condition, the homogeneous wave equations admit the separation of the variables, which is also helpful for solving the nonhomogeneous equations. The resultant ordinary differential equation for the radial coordinate gives a natural extension of the Teukolsky equation.

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.

Solvability in the sense of sequences for some linear and nonlinear Fredholm operators with the logarithmic Laplacian

We study the solvability of certain linear and nonlinear nonhomogeneous equations in one dimension involving the logarithmic Laplacian and the transport term. In the linear case we show that the convergence in L 2 ( R ) of their right sides yields the existence and the convergence in L 2 ( R ) of the solutions. We generalize the results obtained in the earlier article of Efendiev and Vougalter [Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian. Monatsh Math. 2023] in the non-Fredholm case without the drift. In the nonlinear part of the work we demonstrate that, under the reasonable technical assumptions, the convergence in L 1 ( R ) of the integral kernels implies the existence and the convergence in L 2 ( R ) of the solutions.

Numerical analysis of air–water two-phase upflow in artificial upwelling of deep ocean water by airlift pump

Artificial upwelling by the use of airlift pump is regarded as an effective way in utilizing deep ocean water and actualizing ocean fertilization. This paper focuses on numerical analysis of steady air–water two-phase flow in a vertical pipe of an airlift system based on one-dimensional multi-fluid model. The depth distributions of 6 physical quantities such as volumetric fractions and axial velocities of two phases, air density and pressure are calculated by solving the governing equations or by integrating the vector form of nonhomogeneous ordinary differential equation for two-phase flow interval. Upon successful verification of the present numerical model through a comparison with precedent theoretical and experimental results in case of a vertical pipe with length of 7.86 m, the model is extended to the case of artificial upwelling from water depth of 2000 m. The effects of submerged depth of air–water mixer on the pumped amount of water and the depth distributions of 6 physical quantities are considered.