Nonhomogeneous Equations Research Articles

Multiple solutions for a singular nonhomogenous biharmonic equation in Heisenberg group

In this paper, we study a singular nonhomogenous biharmonic problem with Dirichlet boundary condition in Heisenberg group$ \begin{equation*} \left \{\begin{array}{ll} \Delta^2_{{\mathbb{H}^1}}u = \frac{f(\xi, u)}{\rho(\xi)^\beta}+\epsilon h(\xi)\ &\mbox{in} \ \Omega, \\ u = \frac{\partial u}{\partial \nu} = 0, \ \ &\mbox{on}\ \partial\Omega, \end{array} \right. \end{equation*} $where $ \Omega\subset {\mathbb{H}^1} $ is a bounded smooth domain, $ \Delta^2_{{\mathbb{H}^1}}u = \Delta_{\mathbb{H}^1}(\Delta_{\mathbb{H}^1}u) $ denotes the biharmonic operator in Heisenberg group $ \mathbb{H}^1 = \mathbb{C} \times \mathbb{R} $, $ 0\leq \beta < 4 $ with $ 4 $ is the homogeneous dimension of $ {\mathbb{H}^1} $ and $ f:\Omega \times \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function which satisfies subcritical and critical exponential growth condition, $ h(\xi)\in (D_0^{2, 2}(\Omega))^* $, $ h(\xi)\geq0 $ and $ h(\xi) \not\equiv0 $, $ \rho(\xi) = (|z|^4+t^2)^{\frac{1}{4}} $, $ \xi = (z, t)\in \mathbb{H}^1 $ with $ z = (x, y)\in\mathbb{R}^{2} $, $ \epsilon $ is a small positive parameter. We obtain the existence and multiplicity of solutions by the Ekeland variational principle, mountain pass theorem and singular Adams inequality in Heisenberg group.

A Study of a Neutrosophic Differential Equation by Using the One-Dimensional Geometric AH-Isometry

In this paper, the definition of a Neutrosophic Differential Equation by Using the One-Dimensional Geometric AH-Isometry. The main objective is define a Neutrosophic identical linear differential equation and Neutrosophic non-homogeneous linear differential equation and find solutions for this equation.

Gradient-Based Differential Neural-Solution to Time-Dependent Nonlinear Optimization

In this technical article, to seek the optimal solution to time-dependent nonlinear optimization subject to linear inequality and equality constraints (TDNO-IEC), the gradient-based differential neural-solution, termed as GDN model, is proposed and researched. Notably, TDNO-IEC is first converted into the nonhomogeneous linear equation with the dynamic parameter. Second, differential neural-solution with the aid of gradient is designed. The contrastive theoretical analyses among the GDN model, gradient-based neural network (GNN), and the dual neural network (DNN) prove that the proposed GDN model has higher accuracy for eliminating the large solution error with exponential convergence. In addition, reasonable convergent time of the GDN model is guaranteed by activation functions with simple formulation. Last, an illustrative example and real-world applications, including robot motion planning and data dimension reduction and reconstruction, further validate the high availability of the proposed GDN model.

Non-asymptotic Disturbances Estimation for Time Fractional Advection-Dispersion Equation

In this paper, a method for disturbance estimation for a class of non-homogeneous time fractional advection dispersion equation with general boundary conditions, is proposed. The disturbances are considered to affect the boundary conditions or the measurements or both. The proposed method is based on modulating functions. Numerical simulations are presented to illustrate the performance of the approach.

AN ANALYTICAL STUDY OF SPACE-TIME FRACTIONAL ORDER GAS DYNAMIC EQUATIONS

In this article, the Sumudu transform with iterative method is implemented to obtain approximate analytical solutions in series form to non-linear homogeneous and non-homogeneous space-time fractional gas dynamic equations. The fractional derivatives presented here are in the Caputo sense. Furthermore, the findings of this study are graphically represented and the solution graphs demonstrate a strong connection between the approximate and exact solutions.

A tripled coincidence point technique for solving integral equations via an upper class of type II

<abstract><p>The goal of this paper is to obtain some tripled coincidence point results for generalized contraction mappings in the setting of $ JS $-metric spaces endowed with a partial order. Furthermore, illustrative examples to support the theoretical results and the application are obtained. Finally, some theoretical results are applied to discuss the existence of a solution for a system of non-homogeneous and homogeneous integral equations as applications.</p></abstract>

Hölder continuity of solutions for a class of drift-diffusion equations

<p style='text-indent:20px;'>We provide several regularity results for non-homogeneous drift-diffusion equations with applications to general dissipative SQG. Our results unify in a rather simple way several previously known results. We build the estimates on an algebraic identity (for the refined energy argument) which relates any integral operator with pure powers of the laplacian.</p>

On the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations in terms of (p,q)-order

On the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations in terms of (p,q)-order

On Some Results About the Asymptotic Behaviour for Some Non-Homogeneous and Non-Linear Differential Equations with A Laplacian p≥1

In this paper, we study the global existence and the asymptotic behaviour for the solutions of the following non-linear and non-homogeneous differential equation with order 3 and with a Laplacian p≥1 〖〖〖〖[|u〗^(ˊˊ) (t)|〗^(p-1) u〗^(ˊˊ) (t)]〗^ˊ+f(t,u(u))=e(t) ;p≥1 Where t→+∞. Also, we get the solutions of (1), which the type of the asymptotic behaviour of the global solutions is at^2+bt+c; t→+∞, a,b,c ∈R ;a ≠0

Modified Variational Iteration Method Approach for Fuzzy Initial Value Problem

In this paper, an application of  modified variational iteration method (MVIM) for solving homogeneous and nonhomogeneous ordinary differential equations with fuzzy initial value conditions (FDEs) are examined. This modified form is about a procedure some changes in the general formula of the variational iteration method (VIM) . MVIM showed a qualitative and excellent improvement over the standard VIM. To illustrate how this modified form is works we picked some numerical examples and solved them. The efficacy and validity of MVIM have been confirmed through results obtained.  

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method (LKM) with the dual reciprocity method (DRM). Firstly, the temporal derivative is discretized by a finite difference scheme, and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation. Secondly, the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution. And then, the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation, respectively. The LKM is a recently proposed local radial basis function collocation method with the merits of being simple, accurate, and free of mesh and integration. Compared with the traditional domain-type and boundary-type schemes, the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains. Numerical experiments, including two- and three-dimensional heat transfer models, demonstrated the effectiveness and accuracy of the new methodology.

Direct methods for singular integral equations and non-homogeneous parabolic PDEs

In this article, the author presented some applications of the Laplace, \(L^2\), and Post-Widder transforms for solving fractional Singular Integral Equations, impulsive differential equation and systems of differential equations. Finally, analytic solution for a non-homogeneous partial differential equation with non-constant coefficients is given. The obtained results reveal that the integral transform method is an effective tool and convenient.

Lotka-Volterra Model with Periodic Harvesting

A closed interaction of predator prey is considered. The interaction is expressed in the Lotka-Volterra model. Two types of Lotka-Volterra models are considered, with and without carrying capacity of the prey. The paper includes a periodic harvesting of predator and/or prey, a function of time which acts to the model. Hence, the model is in the form of a system of non-homogeneous equations. Dynamical properties of the models are investigated. The solutions are computed numerically. Such interaction is in the need of integrated farming on harvesting of predator and/or prey. In this model the number of population in the system is sensitive to the initial value, which can be applied to the integrated farming systems such that the system remains sustainable.

متسلسلات هار النسبية لتقريب معادلات الحالة المتجانسة وغير المتجانسة

In this paper, we give a new approximate solution method to homogeneous and non-homogeneous state equations using rationalized Haar functions. Integration operational matrix of rationalized Haar functions are used to convert the computation of homogeneous and non homogeneous state equations to a simple system of algebraic equations. By using the method (based on Matlab programming) on numerical analysis examples, we show that our method has high degree of accuracy.

On the Stability of the van der Pol-Mathieu-Duffing Oscillator under the Effect of Fast Harmonic Excitation

This paper aims to examine the nonlinear dynamics of a van der Pol-Mathieu-Duffing oscillator under the effect of fast harmonic excitation. The governing equations of motion describing the harmonically forced oscillations of the van der-Pol-Mathieu-Duffing oscillator are expressed in terms of the second-order nonhomogeneous nonlinear ordinary differential equation with suitable initial conditions. This paper uses Krylov-Bogoliubov averaging technique for the stability analysis of the system. The frequency response curves under the effect of external excitation, damping, and nonlinearity are obtained at various resonances. Additionally, the stable and unstable regions were identified. It turns out that the damping reduces the amplitude of oscillations and squeezes the instability regions, whereas the stability region grew with the increase in the amplitude of external excitation.

Approximate Solution of Two Dimensional Disc-like Systems by One Dimensional Reduction: An Approach through the Green Function Formalism Using the Finite Elements Method

We present a comprehensive study for common second order PDE’s in two dimensional disc-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on the formalism, we propose to secure an exact solution via a Fourier expansion of the Green function, which entails solving an infinitely countable system of differential equations for the Green–Fourier modes that in the simplest case yields the source-free Green distribution. We present results on non separable systems—or such whose solution cannot be obtained by the usual variable separation technique—on both annulus and disc geometries, and show how the resulting one dimensional Fourier modes potentially generate a near-exact solution. Numerical solutions will be obtained via finite differentiation using Finite Difference Method (FDM) or Finite Element Method (FEM) with the three-point stencil approximation to derivatives. Comparing to known exact solutions, our results achieve an estimated numerical relative error below 10−6. Solutions show the well-known presence of peaks when r=r′ and a smooth behavior otherwise, for differential equations involving well-behaved functions. We also verified how the Green functions are symmetric under the presence of a “weight function”, which is guaranteed to exist in the presence of a curl-free vector field. Solutions of non-homogeneous differential equations are also shown using the Green formalism and showing consistent results.

New Analytical Method For Solution of Second Order Ordinary Differential Equations With Variable Coefficients

In this study, we propose four methods for the analytical solution of second order ordinary non-homogeneous differential equations with variable coefficients. In the first case, the method directly gives the solution in explicit or integral form. In the second method, the solution of the problem reduces to the solutions of adjoined second order ordinary differential equation of homogeneous type. As long as the analytical solution of two adjoined equations can be solved, the analytical solution can always be found. In the third method, the differential equation is transformed into Riccati equation. Riccati equation is solved by means of a method recently developed. In order to solve non-homogeneous differential equation, the fourth method is developed. The strategy is different but the solution is again based on the solution of Riccati equation.

On bounded solutions of linear systems of differential equations with unbounded coefficients

This paper deals with a problem of finding a bounded solution of a system of nonhomogeneous linear differential equations with an unbounded matrix of coefficients on a finite interval. The right-hand side of the equation belongs to a space of continuous functions bounded with some weight; the weight function is chosen taking into account the behavior of the coefficient matrix. The problem is studied using a modified version of the parameterization method with non-uniform partitioning. Necessary and sufficient conditions of well-posedness of the problem are obtained in terms of a bilaterally infinite matrix of special structure.

Molecular Photothermal Effects on Time-Resolved IR Spectroscopy: Solute-Solvent Intermolecular Energy Transfer.

Time-resolved IR pump-probe (IR-PP) and two-dimensional IR (2D-IR) spectroscopy are valuable tools for studying ultrafast chemical and biological processes in solutions. However, the corresponding signals at long times are obscured by the molecular photothermal effects resulting from the heat dissipation of vibrationally photoexcited molecules to the surroundings. Recently, a phenomenology model was used to describe molecular photothermal effects on IR-PP signals and the diagonal and cross-peaks of 2D-IR spectra at long pump-probe delay times. Here, we consider the thermal diffusion equation with a time-dependent heat source term to describe the solute-solvent energy transfer process. An approximate solution to the nonhomogeneous differential equation shows that the molecular photothermal effect is determined by the mean intermolecular distance between IR-absorbing molecules. We show that the time profile of heat dissipation from a vibrationally excited molecule to the surroundings, which provides information about the mechanisms involved in the solute-solvent intermolecular energy transfer process in solutions, can be directly measured by analyzing the molecular photothermal IR-PP and 2D-IR signals. We anticipate that the present work can be used to interpret local heating-induced time-resolved IR spectroscopic signals and understand the rate of and the mechanisms involved in the conversion from high-frequency molecular vibrational energy to solvent kinetic energy in condensed phases.

Strong Maximum Principle and Boundary Estimates for Nonhomogeneous Elliptic Equations

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear nonhomogeneous degenerate elliptic equations on the formF(x,u,Du,D2u)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ F(x,u,Du,D^{2}u) = 0 $$\\end{document} under suitable assumptions allowing for non-Lipschitz growth in the gradient term. In case of smooth boundaries, we also prove a Hopf lemma, a boundary Harnack inequality, and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply, e.g., to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.