Nonhomogeneous Differential Equations Research Articles

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

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.

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.

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.

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.

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.

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.

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.

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.

ON THE BEHAVIORS OF SOLUTIONS IN LINEAR NONHOMOGENEOUS DELAY DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

This paper deals with the behaviors of solutions for linear nonhomogeneous delay differential equations. In this study, a periodic solution, an asymptotic result and a useful exponential estimate of the solutions are established. Our results are obtained by the use of real roots of the corresponding characteristic equation.

Exact higher-order moments for linear non-homogeneous stochastic differential equation

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

Backstepping model‐free adaptive control for a class of second‐order nonlinear systems

SummaryThis paper proposes a backstepping model‐free adaptive control (BS‐MFAC) algorithm for the trajectory tracking control problem of a class of second‐order nonlinear systems. Firstly, the model‐free adaptive control (MFAC) is employed to track the virtual desired velocity. Then, based on the measured state data at the current moment, a virtual desired velocity is designed, combining MFAC with backstepping control in a clever manner. By designing a discrete form of Lyapunov function, it is proven that the system velocity error under this algorithm converges asymptotically and stably. Furthermore, a nonhomogeneous difference equation solution related to the system position is obtained through calculations. The analysis of the solution indicates that the system position output error will slide into a neighborhood around zero. This algorithm possesses advantages such as simple structure, model independence, and strong robustness. Additionally, a planar two‐link space robot arm model is constructed using the Mbdyn platform, a multibody dynamics simulation software. Trajectory tracking control of joint angles is achieved under the proposed control algorithm in this paper. The simulation outcomes unequivocally establish the efficacy of the algorithm, which adeptly exploits the pseudo partial derivatives (PPD) estimation for second‐order nonlinear systems in MFAC. The algorithm yields expedited and superior trajectory tracking control of said systems.

Theoretical Analysis on Thermo-Mechanical Bending Behavior of Timber–Concrete Composite Beams

In this study, an analytical approach is introduced for predicting the bending behavior of a timber–concrete composite (TCC) beam subjected to a mechanical load and a non-uniform temperature field, in which the orthotropy of timber as well as interfacial slip are taken into consideration. The analytical model addresses the non-uniform temperature field using Fourier series expansion based on the heat transport theory. The stresses and displacements of the TCC beam under the thermo-mechanical condition are governed by the thermo-elasticity theory, and the corresponding solution is derived analytically by solving a group of non-homogeneous partial differential equations. The proposed solution is in good agreement with the finite element solution and exhibits higher accuracy compared to the Euler–Bernoulli beam solution that relies on the assumption of transverse shear deformation and isotropy. An extensive investigation is carried out to analyze how the bending behavior of TCC beams is influenced by variations in interfacial shear stiffness and temperature field.

The Elzaki transform and its applications for Hyers -Ulam stability of higher order differential equations using Covid 19 method

The Elzaki transform is being used to present the generic solution and prove the Hyers-Ulam stability of higher-order linear differential equations. In addition, the stability of homogeneous and non-homogeneous linear differential equations is discussed using the Mittag-Leffler-Hyers-Ulam method. Furthermore, such equations’ stability constants are determined.

Multistability, period-adding, and fractality in a plasma oscillator

In this paper, we report on a periodically driven plasma oscillator modeled by a six-parameter nonhomogeneous second-order ordinary differential equation. We fix four of these parameters, and investigate the dynamics of this system by varying the other two, namely, the amplitude A and the angular frequency ω of the driving. In other words, we investigate the (ω,A) parameter plane, where the dynamical behavior of each point was characterized by the magnitude of the largest Lyapunov exponent. Then, we show that this parameter plane reveals the occurrence of the multistability phenomenon in the system. Properly generated bifurcation diagrams confirm this finding. Basins of attraction of coexisting periodic and chaotic attractors in the phase-space are presented. We also report on the organization of periodicity and chaos in the (ω,A) parameter plane. Typical periodic structures were detected embedded in a chaotic region, namely, the cuspidal, the non-cuspidal, and the shrimp-like. At a certain location on the parameter plane, the organization of the shrimp-like periodic structures resembles a fractal, since the same shape is seen when we look through different scales. Elsewhere these same structures appear organized in a period-adding sequence.

A liquid-solid-chemical coupled mass transport model for concrete considering phase assemblages and microstructure evolution

Concrete is a highly alkaline porous medium that is prone to chemical reactions with acidic environments. The ion transport coupled with chemical reactions between concrete and solutions in acidic environments (acid rain, acid groundwater, acid industrial water, etc.) is a major challenge to the durability design of such infrastructure. Here, a liquid–solid-chemical coupled mass transport model of concrete considering phase assemblages and microstructure evolution was developed to tackle this problem. The highly nonlinear evolution of phase assemblages was solved via chemical thermodynamics. Further analysis was conducted on the spatiotemporal relationship of microstructure evolution caused by reaction kinetics, and the dynamic porosity and ion adsorption capacity were taken into account in the mass transfer equations. Furthermore, the nonlinear non-homogeneous partial differential equations were solved via finite difference method. Finally, the proposed model was validated based on four different types of cement-based concrete. The results indicated that considering the chemical reaction between highly alkaline concrete and acidic environments was crucial for the transport of chloride ions. The migration depth varied up to twice under the influence of acidic solutions. This study builds a bridge between mass transfer and cement chemistry knowledge, and is expected to improve the accuracy of life prediction for reinforced concrete structures.