Nonhomogeneous Equations Research Articles

Oscillation of arbitrary-order derivatives of solutions to the higher order non-homogeneous linear differential equations taking small functions in the unit disc

<abstract><p>In this article, we study the relationship between solutions and their arbitrary-order derivatives of the higher order non-homogeneous linear differential equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>in the unit disc $ \bigtriangleup $ with analytic or meromorphic coefficients of finite $ [p, q] $-order. We obtain some oscillation theorems for $ f^{(j)}(z)-\varphi(z) $, where $ f $ is a solution and $ \varphi(z) $ is a small function.</p></abstract>

The growth of solutions of non-homogeneous linear differential equations

The growth of solutions of non-homogeneous linear differential equations

A semi-analytical method for solving a class of non-homogeneous time-fractional partial differential equations

A semi-analytical method for solving a class of non-homogeneous time-fractional partial differential equations

Null controllability for a singular heat equation with a memory term

In this paper we focus on the null controllability problem for the heat equation with the so-called inverse square potential and a memory term. To this aim, we first establish the null controllability for a nonhomogeneous singular heat equation by a new Carleman inequality with weights which do not blow up at t = 0. Then the null controllability property is proved for the singular heat equation with memory under a condition on the kernel, by means of Kakutani’s fixed-point theorem.

First-order linear differential equations whose data are complex random variables: Probabilistic solution and stability analysis via densities

<abstract><p>Random initial value problems to non-homogeneous first-order linear differential equations with complex coefficients are probabilistically solved by computing the first probability density of the solution. For the sake of generality, coefficients and initial condition are assumed to be absolutely continuous complex random variables with an arbitrary joint probability density function. The probability of stability, as well as the density of the equilibrium point, are explicitly determined. The Random Variable Transformation technique is extensively utilized to conduct the overall analysis. Several examples are included to illustrate all the theoretical findings.</p></abstract>

Dean Vortex for Laminar Flows in Curved Pipes with Various Cross-Sections

Curved pipes flows are encountered in different areas such as heat transfer, chaotic mixing, separation of mixtures in pipes, or blood circulation among others and exhibit a variety of characteristics depending on the ranges of Dean numbers and pipe curvatures. Studies on curved pipes flows usually consider the cases of circular, elliptical, and rectangular shapes for the cross sections of the pipe. The present work extends the availability of asymptotic analytical solutions to new ranges of cross-sectional shapes while considering fully developed steady state flows at low Dean numbers. The new shapes are given by a polar equation R* (q) satisfying the relation 1-R^(*2) (q)+dR^(*y) (q)sin(yq)=0 where d and y are parameters. The zero-order azimuthal velocity profiles for various cross-sections are given by exact analytical solutions. Solutions for the nonhomogeneous biharmonic equation for the secondary flows are given by using exact expressions for the particular solutions. Furthermore, the Fourier series decomposition of the solution is adopted to determine the integration constants that allow satisfying the non-slip boundary conditions. Solutions are presented for semi triangular (y=3) , square (y=4), and pentagonal (y=5) cross sections shapes. It is found that the velocity distribution and the Dean’s vortexes intensities are modified in function of the cross-section shapes.

Development of thermal model for the determination of SLM process parameters

In this study, a mathematical model is developed for determining the process parameters used in the manufacturing process of powder materials with selective laser melting method. Although the studies carried out to date cover detailed modeling studies including three-dimensional and time-dependent situations, obtaining quite different approaches from experimental results has led to the idea that a change should be made in the construction of the mathematical problem. Therefore, different from other studies and for the first time, the Eigen-function expansion method was used in the analytical solution of the selective laser melting thermal model. The developed mathematical model includes the steady-state solution with the appropriate boundary conditions of the 2-D non-homogeneous heat equation. The mathematical model was first solved analytically with the Eigen-function expansion method, and the function obtained as a result of the solution was introduced into the MATLAB software. The parametric study was performed numerically over laser power, laser spot size and powder bed thickness. With the developed model, the solution was converged in 30 s and 11.5% more accurate with respect to experimental results were obtained in width, 29% in depth and 3% in temperature as compared to other analytical models exist in the literature.

Laplace Differential Transform Method for Solving Nonlinear Nonhomogeneous Partial Differential Equations

In this paper, the Laplace Differential Transform Method (LDTM) was utilized to solve some nonlinear nonhomogeneous partial differential equations. This technique is the combined form of the Laplace transform method with the Differential Transform Method (DTM). The combined method is efficient in handling nonlinear nonhomogeneous partial differential equations with variable coefficients. Laplace transform is introduced to overcome the inadequacy resulted from unsatisfied boundary condition in using DTM. Illustrative examples were examined to demonstrate the effectiveness of Laplace differential transform method. Results revealed that the LDTM is well appropriate for use in solving such problems.

A systematic review of water resources in Yellow River Basin of China

This paper reviews the literature on the Yellow River Basin’s water resources from 2000 to 2019 based on the Web of Science. The annual publication, source country and source journal are analyzed using SATI4.0. The nonhomogeneous linear difference equation, comprehensive score analysis and Bradford’s law are used for fundamental analysis. CiteSpace is used for co-occurrence analysis and burst detection of noun phrases. Finally, Carrot2 is used to cluster the literature to get a knowledge framework for the research. Results showed that: (1) The research process of water resources in the Yellow River Basin (YRB) can be divided into three stages. This topic will continue to flourish in the future. (2) The research covers a wide range of contents, which mainly includes runoff changes, runoff-sediment relationship, water resources allocation, disaster response, the impact of human activities and climate change, as well as ecological protection and sustainable development. Human activities and the impact of climate change are the focus of research. (3) In the future, scholars should adhere to the belief of green and sustainable development of YRB and focus on the study of the Loess Plateau.

New Method of Solving the Economic Complex Systems

In this study, the authors first develop a direct method used to solve the linear nonhomogeneous time-invariant difference equation with the same number for inputs and outputs. Economic cybernetics is the crystallization for the integration of economics and cybernetics. It analyzes the stability, controllability, and observability of the economic system by establishing a system model and enables people to better understand the characteristics of the economic system and solve economic optimization problems. The economic model generally applies the discrete recurrence difference equation. The significant analytic approach for the difference equation is the z-transformation technique. The z-transformation state of the economic cybernetics state-space difference equation generally is a rational function with the same power for the numerator and the denominator. The proposed approach will take the place of the traditional methods without all annoying procedures involving the long division of some complicated polynomials, the expanded multiplication of many polynomial factors, the differentiation of some complicated polynomials, and the complex derivations of all partial fraction parameters. To highlight the novelty of this research, this study especially applies the proposed theorems originally belonging to engineering to the field of economic applications.

Positive maximal and minimal solutions for non-homogeneous elliptic equations depending on the gradient

We are concerned with positive maximal and minimal solutions for non-homogeneous elliptic equations of the form−div(a(|∇u|p)|∇u|p−2∇u)=f(x,u,∇u) in Ω, supplied with Dirichlet boundary conditions. First we localize maximal and minimal solutions between not necessarily bounded sub-super solutions. Then using a uniform gradient estimate, which seems of independent interest, we show the existence of positive maximal and minimal solutions in some situations. More precisely, we obtain positive maximal and minimal solution to some classes of non-homogeneous equations depending on the gradient which may be perturbed by unbounded, singular or logistic sources.

Equivalence of weak and viscosity solutions in fractional non-homogeneous problems

We establish the equivalence between the notions of weak and viscosity solutions for non-homogeneous equations whose main operator is the fractional p-Laplacian and the lower order term depends on x, u and $$D_s^p u$$ , being the last one a type of fractional derivative.

Cubic spline fractal solutions of two-point boundary value problems with a non-homogeneous nowhere differentiable term

Fractal interpolation functions (FIFs) supplement and subsume all classical interpolants. The major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function. This work proposes the use of cubic spline FIFs through moments for the solutions of a two-point boundary value problem (BVP) involving a complicated non-smooth function in the non-homogeneous second order differential equation. In particular, we have taken a second order linear BVP: y′′(x)+Q(x)y′(x)+P(x)y(x)=R(x) with the Dirichlet’s boundary conditions, where P(x) and Q(x) are smooth, but R(x) may be a continuous nowhere differentiable function. Using the discretized version of the differential equation, the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function. These moments are then used to construct the cubic fractal spline solution of the BVP, where the non-smooth nature of y′′ can be captured by fractal methodology. When the scaling factors associated with the fractal spline are taken as zero, the fractal solution reduces to the classical cubic spline solution of the BVP. We prove that the proposed method is convergent based on its truncation error analysis at grid points. Numerical examples are given to support the advantage of the fractal methodology.

On the Complex Oscillation of Meromorphic Solutions of Nonhomogeneous Linear Differential Equations with Meromorphic Coefficients

In this paper, we study the higher order differential equation f k + B f = H , where B is a rational function, having a pole at ∞ of order n > 0 , and H ≡ 0 is a meromorphic function with finite order, and obtain some properties related to the order and zeros of its meromorphic solutions.

Transient thermoelastic analysis of a semi-infinite circular plate with the power-law nonhomogeneity subjected to ramp-type sectional heating

The main objective is to investigate the transient thermoelastic response on a nonhomogeneous semi-infinite elastic circular plate that is subjected to ramp-type heating over the concentric circular region of the upper face. The nonhomogeneous plate has thermo-mechanical properties expressed in the form of power-law through the plate thickness with variable z and with constant Poisson's ratio. The transient nonhomogeneous heat conduction differential equation is solved using a modified integral transformation technique in terms of modified Bessel functions. The associated thermal stress components are evaluated theoretically by incorporating the thermoelastic displacement potential feature based on the Boussinesq-Papkovich method. Numerical calculation is carried out the functionally graded ceramic – metal based material plate, in which partially stabilized zirconia (PSZ) and austenitic stainless steel (SUS304) are subjected to ramp-type heating on one of the plane boundaries; whereas the other plane boundary is held at zero temperature. The analyzed findings indicate strong consent with those available in the literature.

An Eigen Expansion Method for Elastic Boundary Conditions

With the aid of symplectic system, the mechanical problems related to elastic plates are studied. Using this method, displacement and stress components are taken as basic variables, so it is no longer necessary to make assumptions like the semi inverse method, and thus a set of concrete methods for dealing with non-homogeneous equations and boundary conditions are given. The mechanical response of elastic plates under external forces is discussed.

Homogeneous and Non Homogeneous Ordinary Differential Equations with the Second Order

Since Newton time, Differential Equations have been used to comprehend most of physical, geometrical, and vital science in addition to their participation in the study of arithmetic analysis. Furthermore, they have been used in economical and social aspects. They have been developed and got significant position in various sciences. This paper presents for the non-homogeneous ordinary differential equations with the second order. This idea starts in chapter one which talks about the notion of those equations, their orders, in addition to the study of the linear differential equation with the first order and its solution. Whereas the second chapter studies the ordinary differential equations (homogeneous and non-homogeneous) and their solution. Each chapter is supported by solved examples that cover most of the aspects of the topic and we hope that this paper will make a good reference for those who would like to study this topic furtherly.

Symplectic Solutions in Stress Analysis of Thin Plates

Aiming at the common elastic thin plate materials in engineering, the dual control differential equation and the complete solution space composed of zero eigensolution and non-zero eigensolution are established in the state space. On this basis, the non-homogeneous condition of the boundary and the solution method of the non-homogeneous equation are studied systematically. The stress distribution of thin plate bending problem is discussed with an example, and the stress concentration of material under special constraints is analyzed.

Numerical solutions of the initial boundary value problem for the perturbed conformable time Korteweg‐de Vries equation by using the finite element method

AbstractIn this paper, we investigate the initial‐boundary‐value problem for the nonhomogeneous Korteweg‐de Vries equation with conformable derivative on time part of it. We use the finite element method with B‐spline as the basis functions for obtaining the numerical solutions for this nonlinear equation. In addition, we prove a posteriori and a priori errors for it. These show the adaptivity and convergence of our method. Also, a posteriori error estimate concludes that the error estimate decreases as α increases. We show the accuracy of our work by comparing with the exact solution for the homogeneous KdV equation. We also bring an example for the nonhomogeneous conformable time KdV equation to demonstrate the accuracy and efficiency of the proposed method. Also, these numerical results are consistent with the result of theorems. The numerical results are given in tables and figures.

On the existence and uniqueness of solution of boundary‐domain integral equations for the Dirichlet problem for the nonhomogeneous heat transfer equation defined on a 2D unbounded domain

A system of boundary‐domain integral equations (BDIEs) is obtained from the Dirichlet problem for the diffusion equation in nonhomogeneous media defined on an exterior two‐dimensional domain. We use a parametrix different from the one employed in Dufera and Mikhailov (2019). The system of BDIEs is formulated in terms of parametrix‐based surface and volume potentials whose mapping properties are analyzed in weighted Sobolev spaces. The system of BDIEs is shown to be equivalent to the original boundary value problem and uniquely solvable in appropriate weighted Sobolev spaces suitable for unbounded domains.