Nonhomogeneous Equations Research Articles

Analytical Approximation of Advertising Impact on Sale of Products Using Differential Equation Model

Advertisement involves significant amount of money and remain as one of the major issues bothering companies and other investment sectors. Advertising is a marketing strategy directed to the consumers through any media in order to present and promote product, services and any other cause. Sale of products occurs with or without advertisement and may increase or decrease depending on the market situation. The question is, is there a mathematical model that can surely confirm the effects of advertising on sales of products? The aim of this paper is to develop a first order linear differential equation model that seeks to justify the impacts of advertising on the sale of products in Ghanaian market. Two basic variables, the advertisement and sales of product were considered for the model. A system of two differential equations from the two basic variables were developed and converted to first order non-homogeneous linear differential equation. The geometrical method was employed to obtain qualitative information about the solution of the first order non-homogeneous linear differential equation. The MATLAB software package was used to plots the graphs to illustrate the behaviour of the sale of products with respect to the time of advertisement. The findings revealed that differential equation model can be used to justify the advertising impacts on the sale of products.

Nonhomogeneous equations with critical exponential growth and lack of compactness

We study the existence and multiplicity of positive solutions for the following class of quasilinear problems \[-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,\] where \(\epsilon\) is a positive parameter. We assume that \(V:\mathbb{R}^N \to \mathbb{R}\) is a continuous potential and \(f:\mathbb{R}\to\mathbb{R}\) is a smooth reaction term with critical exponential growth.

Theory and Experiment of Enclosing Control for Second-Order Multi-Agent Systems

This article investigates the enclosing control without preset formation of second-order multi-agent systems for stationary targets. This article uses a directed graph to describe and the direction of information exchange between agents and targets. For continuous-time systems, an enclosing control algorithm is proposed, which does not need to preset the desired formation. The state transfer equation is used to transform the solution of the system into a matrix function of first-order linear constant-coefficient non-homogeneous differential equations. By analyzing the convergence of the solution, the value range of the gain parameter is obtained, and the requirements of topology are proposed. Then the discrete protocol is applied to the discrete-time system. Based on the Schur stability analysis of the system, the requirements of topology and parameter for the system to achieve enclosing control are given. Finally, the self-designed multi-agent platform is introduced, and simulation and experimental results are presented to validate the effectiveness of the protocol.

Fractional Derivatives of Some Fractional Functions and Their Applications

In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.

Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions

In this paper, we study saturable nonlinear Schrodinger equations with nonzero intensity function which makes the nonlinearity become not superlinear near zero. Using the Nehari manifold and the Lusternik-Schnirelman category, we prove the existence of multiple positive solutions for saturable nonlinear Schrodinger equations with nonzero intensity function which satisfies suitable conditions. The ideas contained here might be useful to obtain multiple positive solutions of the other non-homogeneous nonlinear elliptic equations.

Gradient regularity for a singular parabolic equation in non-divergence form

In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations \begin{document}$ \partial_t u-|Du|^\gamma\Delta_p^N u = f, $\end{document} where $ -1<\gamma<0 $, $ 1<p<\infty $, and $ f $ is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.

Performance Based Time History Analysis of Five Story Shear Frame Building Using MATLAB and ETABS

This paper compares the time history analysis results for state space representation (SSR) method using MATLAB with the result conducted using ETABS. The equation of motion of the structure subjected to seismic excitation represented by second-order linear non-homogeneous differential equation. This equation reduced to two coupled first order differential equations and state space representation was formulated to represent the system in matrix form and MATLAB Simulink was used to determine response of the structure. The objectives of this study are i) to conduct a comparative study between the state space representation which is a powerful tool with results of ETABS. ii) to investigate the accuracy of SSR method. iii) to conduct a performance based dynamic analysis for shear frame structures and study outcome responses of the structure. This analysis was based on the assumptions, i) the total story mass is lumped at the center of story diaphragm. ii) No deflection occurs in beams; story beams are infinitely rigid in comparison to story columns. iii) no changes in the nature of the boundary conditions during and after the analysis iv) the system is elastic linear time-invariant (ELTI) and material nonlinearity is not considered. So that that structural degree of freedom decreased to be equivalent to the number of storys. The results showed a significant similarity in comparison with ETABS software. The maximum absolute difference of displacement and story drift ratio was 3.35mm and 0.0016 was obtained at the roof of third and fifth story respectively.

HOMEOMORPHISMS RELATED TO THE POLYNOMIAL-LIKE ITERATIVE EQUATION ON <inline-formula><tex-math id="M1">$\mathbb{S}^1$</tex-math></inline-formula>

In this paper we study all homeomorphisms on the unit circle $\mathbb{S}^1$, whose lifts are $C^0$ solutions of a class of nonhomogeneous polynomial-like iterative equation. By an auxiliary equation, we present all those homeomorphisms and illustrate our results by examples.

Unsteady Plasma Flow Near an Oscillating Rigid Plane Plate Under the Influence of an Unsteady Nonlinear External Magnetic Field

In an irreversible thermodynamics framework, Stokes’ second problem was examined for unsteady oscillating flow. The objective was to apply that for a plasma near an illimitable oscillating rigid plane plate under the influence of an unsteady nonlinear applied magnetic field. The Bhatnagar-Gross-Krook (BGK) pattern of the Boltzmann kinetic equation supplemented by Maxwell’s equations was investigated. The method of moments was applied with a two-sided distribution function. The exact traveling wave solution was obtained for a system consisting of four non-homogeneous partial differential equations. The velocity, shear stress, viscosity coefficient, generated electric field, applied nonlinear magnetic field, polarization, gyro-radius, and gyro-frequency were calculated. Furthermore, the distinction between the equilibrium velocity distribution function and the perturbed distribution functions was theoretically clarified at distinct time values. The advantage of the Boltzmann equation permitted us to consider irreversible non-equilibrium thermodynamics principles. For that purpose, the calculated distribution functions should be used in the formulae of entropy, entropy flux, thermodynamic forces, and kinetic coefficient. From the analysis of the results, it is found that Boltzmann’s H-theorem, thermodynamics laws, and Le Chatelier’s principle were consistent with our model for the whole system. The distinct contributions of the forces exerted on the system modified its internal energies; they were expressed via the Gibbs formula. The results demonstrated that the proposed model is capable of describing the behavior of plasma helium gas in the upper atmosphere ionic belts. Based on the analytical calculations, 3D-Graphics illustrating the physical quantities were drawn to predict their conduct, and the results are deeply discussed.

On the Identification of Source Term in the Heat Equation from Sparse Data

We consider the recovery of a source term $f(x,t)=p(x)q(t)$ for the nonhomogeneous heat equation in $\Omega\times (0,\infty)$ where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary $\partial\Omega$ from overposed lateral data on a sparse subset of $\partial\Omega\times(0,\infty)$. Specifically, we shall require a small finite number $N$ of measurement points on $\partial\Omega$ and prove a uniqueness result; namely the recovery of the pair $(p,q)$ within a given class, by a judicious choice of $N=2$ points. Naturally, with this paucity of overposed data, the problem is severely ill-posed. Nevertheless we shall show that provided the data noise level is low, effective numerical reconstructions may be obtained.

On the integral operators pertaining to a family of incomplete <i>I</i>-functions

This paper introduces a new incomplete I-functions. The incomplete I-function is an extension of the I-function given by Saxena [ 1 ] which is a extension of a familiar Fox's H-function. Next, we find the several interesting classical integral transform of these functions and also find the some basic properties of incomplete I-function. Further, numerous special cases are obtained from our main results among which some are explicitly indicated. Incomplete special functions thus obtained consisting of probability theory has many potential applications which are also presented. Finally, we find the solution of non-homogeneous heat conduction equation in terms of Incomplete I-function.

Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations

<p style='text-indent:20px;'>In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of nonhomogeneous incompressible Navier-Stokes equations. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm, as well as the <inline-formula><tex-math id="M1">\begin{document}$ L^\infty(H^1) $\end{document}</tex-math></inline-formula> norm. It is mentioned that the mathematical validity of the Prandtl boundary layer theory for nonlinear plane parallel flow is generalized to the nonhomogeneous case.

Application of First-Order Differential Equation to Heat Convection in Fluid

Differential equation is very important in science and engineering, because it required the description of some measurable quantities (position, temperature, population, concentration, electrical current, etc.) in mathematical form of ordinary differential equations (ODEs). In this research, we determine heat transferred by convection in fluid problems by first-order ordinary differential equations. So in this research work first we discuss the solution of ordinary homogeneous and non-homogeneous differential equation and then apply the solution of first-order ODEs to heat transferring particularly in heat convection in fluid.

Existence and uniqueness of solution of first order nonlinear difference equation

This paper is devoted to obtaining the existence and uniqueness result of nonhomogeneous first order nonlinear difference equation with nonlocal condition in cone metric space. Banach contraction principle is used to prove the results. Finally an application of the established result is demonstrated.

Robust Global Boundary Vibration Control of Uncertain Timoshenko Beam With Exogenous Disturbances

In this paper, the dynamics solution problem and the boundary control problem for the Timoshenko beam under uncertainties and exogenous disturbances are addressed. The dynamics of the Timoshenko beam are represented by one non-homogenous hyperbolic partial differential equation (PDE), one homogenous hyperbolic PDE, and three ordinary differential equations (ODEs). The authors suggest a method of lines (MOL) for obtaining the dynamics of the Timoshenko beam in the form of the ODE formula instead of the PDE formula. A global sliding mode boundary control (GSMBC) is designed for vibration reduction of the Timoshenko beam influenced by uncertainties, distributed disturbance, displacement boundary disturbance, and rotation boundary disturbance. Chattering phenomena are avoided by using exponential reaching law reinforced by a relay function. Along the time and position axis: a) the boundary displacements and rotation of the Timoshenko beam are converged to equilibrium; b) the distributed vibrations on both displacements and rotation axis of the Timoshenko are attenuated; c) influence of the exogenous disturbances and system parameters uncertainties are compensated. By using the Lyapunov direct approach, exponential convergence and robustness of the closed-loop system are guaranteed. Finally, simulations are carried out to show that the proposed GSMBC-based MOL scheme is effective for vanishing the vibrations of the Timoshenko beam under uncertainties and external disturbances.

Properties of integrals which have the type of derivatives of volume potentials for one ultraparabolic arbitrary order equation

In weighted Hölder spaces it is studied the smoothness of integrals, which have the structure and properties of derivatives of volume potentials which generated by fundamental solutions of the Cauchy problem for one ultraparabolic arbitrary order equation of the Kolmogorov type. The coefficients in this equation depend only on the time variable. Special distances and norms are used for constructing of the weighted Hölder spaces.&#x0D; The results of the paper can be used for establishing of the correct solvability of the Cauchy problem and estimates of solutions of the given non-homogeneous equation in corresponding weighted Hölder spaces.

Solving a nonhomogeneous integral equation with the variable lower limit

Solving a nonhomogeneous integral equation with the variable lower limit

A Lagrangian approach for inhomogeneous Boussinesq equations

This paper investigates the Cauchy problem for inhomogeneous Boussinesq equations with the data in the scaling invariant Besov spaces: . We shall prove under some smallness assumption on the data: , Boussinesq equations with partial viscosity exists a unique global solution, where we extend recent results as regards the conditions for uniqueness. Using Lagrangian coordinates enables us to solve the nonhomogeneous Boussinesq equations by means of the contraction mapping theorem.

Existence of solutions for a nonhomogeneous semilinear elliptic equation

For a bounded domain Ω, a bounded Carathéodory function g in Ω×R, p>1 and a nonnegative locally integrable function h in Ω which is strictly positive in a set of positive measure we prove that, contrarily with the case h≡0, there exists a solution of the semilinear elliptic problem −Δu=λu+g(x,u)−h|u|p−1u+f,inΩu=0,on∂Ω, for every λ∈R and f∈L2(Ω).

Three-dimensional analytical solution of arbitrarily supported cylindrical panels with weak interfaces using the extended Kantorovich method

This is the first time, a three-dimensional (3D) analytical solution of arbitrarily supported cross-ply cylindrical panel with weak interfaces is developed using multi-term extended Kantorovich method (EKM). A generalized variational based mathematical model is developed by considering the Reissner-type mixed variational principle in conjunction with weak interface model. A linear spring layer model is implemented for depicting the behaviour of weak interfaces. Recently developed EKM is applied to obtain two sets of non-homogeneous ordinary differential equations (ODEs) along radial and circumferential directions, respectively. A suitable solution algorithm is developed for the present case. The accuracy of the presently developed solution is verified by comparing with 3D finite element model using ABAQUS. The multi-term EKM solution is efficient as just two terms yield the accurate results. Further, displacement and stress variations in a laminate at the edges are obtained and some new benchmark results are also presented for arbitrarily supported cylindrical panels with weak interfaces. Effect of the weak interface parameter and other characteristic parameters are investigated to reflect their design aspect. This development may help the researchers to develop analytical models for viscoelastic cases.