Equations Of Kind Research Articles

Existence of Multiple Positive Solutions for Choquard Equation with Perturbation

This paper is concerned with the following Choquard equation with perturbation:-Δu+V(x)u=(1/|x|α∗|u|p)|u|p-2u+g(x),u∈H1(RN), whereN≥3,α∈(0,N), and2-(α/N)<p<(2N-α)/(N-2). This kind of equations is well known as the Choquard or nonlinear Schrödinger-Newton equation. Under some assumptions for the functionsV(x), we prove the existence of multiple positive solutions of the equation. Moreover, we also show that these results still hold for more generalized Choquard equation with perturbation.

Power Series Solution for Solving Nonlinear Burgers-Type Equations

Power series solution method has been traditionally used to solve ordinary and partial linear differential equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-type differential equations in order to demonstrate its scope and applicability.

Reflected backward stochastic differential equations driven by countable Brownian motions with continuous coefficients

In this note, we study one-dimensional reflected backward stochastic differential equations (RBSDEs) driven by Countable Brownian Motions with one continuous barrier and continuous generators. Via a comparison theorem, we provide the existence of a minimal and a maximal solution to this kind of equations.

A NEW DEFY FOR ITERATION METHODS

The work presents an adaptation of iteration method for solving a class of thirst order partial nonlinear differential equation with mixed derivatives.The class of partial differential equations present here is not solvable with neither the method of Green function, the most usual iteration methods for instance variational iteration method, homotopy perturbation method and Adomian decomposition method, nor integral transform for instance Laplace,Sumudu, Fourier and Mellin transform. We presented the stability and convergence of the used method for solving this class of nonlinear chaotic equations.Using the proposed method, we obtained exact solutions to this kind of equations.

Global classical solutions for quantum kinetic fokker-planck equations

We consider a class of nonlinear kinetic Fokker-Planck equations modeling quantum particles which obey the Bose-Einstein and Fermi-Dirac statistics, respectively. We establish the existence and convergence rate to the steady state of global classical solution to such kind of equations around the steady state.

Uncertain fractional differential equations and an interest rate model

The concept of uncertain fractional differential equation is introduced, and solutions of several uncertain fractional differential equations are presented. This kind of equation is a counterpart of stochastic fractional differential equation. By the proposed concept, an interest rate model is considered, and the price of a zero‐coupon bond is obtained. Copyright © 2014 John Wiley & Sons, Ltd.

Formulating grip strength and key pinch strength prediction models for Taiwanese: a comparison between stepwise regression and artificial neural networks

Forceful exertions of the hand have been recognized as a risk factor for the development of work-related musculoskeletal disorders (MSDs) of the upper extremity. Gripping and pinching are frequently used hand strength in various occupational activities and in clinical evaluation of the hand. Therefore, formulating grip and pinch prediction models with easily obtainable personal parameters will help facilitate the design and evaluation of workplace environments or facilitate the hand impairment or progression assessments. This study developed maximum voluntary contraction (MVC) grip and key pinch strength prediction models using regression method and artificial neural networks (ANN). Thirty-three right-handed Taiwanese (19 males and 14 females) voluntarily participated in this study. Grip regression model (adjusted R 2 = 0.897) was formulated based on gender, hand length, weight and thumb length. Key pinch regression model (adjusted R 2 = 0.845) was developed using gender, fingertip to root digit 5 (little finger length), weight and maximum hand breadth. The ANN models predict grip and key pinch strength more accurately than regression models with smaller root mean square errors (RMSEs). However, no significant differences were found between values predicted by both kinds of equations.

Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non-Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Levy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non-Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.

A steady state simulation method for natural gas pressure-relieving systems

The natural gas pressure-relieving system is essential to natural gas pipelines and equipment for their safe operations against overpressure. This paper divided the system into pipe and resistance elements (valve, bent, tee et al.). The pipe elements are simulated by the hydrodynamic and thermodynamic equations, and the resistance elements are simulated by their characteristic equations. Combined those two kinds of equations, as well as the composition-dependent physical property models of natural gas, a steady-state simulation model of pressure-relieving systems is proposed for pressure, flow rate and temperature predictions. The homogeneous assumption is introduced to handle two-phase flow. A velocity constraint is also integrated into the model, making it applicable for both critical and subcritical pressure relief situations. Besides, the proposed solution method is applicable for any pressure-relieving systems composed of any numbers of pipe and resistance elements, as long as there are no branches. Finally, the reliability of the model has been examined using experimental data from three real pressure-relieving systems in Sichuan Province, China. The experiments cover a wide pressure range from 2.23 MPa to 53.80 MPa, and a temperature range from 257.40 K to 335.70 K. The results show that the maximum temperature and flow rate relative errors between the simulation results and experimental data are 1.03% and 3.52%, while the minimum values are 0.22% and 0.31%. The average absolute errors of the temperature and flow rate are 1.83 K and 1.00 × 104 m3/d. The method is also successfully applied to a gas well under critical and subcritical conditions.

Strong and Weak Solutions to Second Order Differential Inclusions Governed by Monotone Operators

In this paper we introduce the concept of a weak solution for second order differential inclusions of the form u″(t) ∈ Au(t) + f(t), where A is a maximal monotone operator in a Hilbert space H. We prove existence and uniqueness of weak solutions to two point boundary value problems associated with such kind of equations. Furthermore, existence of (strong and weak) solutions to the equation above which are bounded on the positive half axis is proved under the optimal condition tf(t) ∈ L1(0, ∞; H), thus solving a long-standing open problem (for details, see our comments in Section 3 of the paper). Our treatment regarding weak solutions is similar to the corresponding theory related to the first order differential inclusions of the form f(t) ∈ u′(t) + Au(t) which has already been well developed.

The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation

It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations.

Biharmonic elliptic problems involving the 2nd Hessian operator

In this paper we will study the equation $$\begin{aligned} \Delta ^2 u=S_2(D^2u),\quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ with \(N=3,\) where \( S_2(D^2u)(x)=\sum _{1\le i<j\le {N}}\lambda _i(x)\lambda _j(x)\), being \(\lambda _i,\) the solutions to the equation $$\begin{aligned} \mathrm{det}\left( \lambda I-D^2u(x)\right) =0, \end{aligned}$$ \(i=1,\dots ,N,\) and \(\Omega \) is a bounded domain with smooth boundary. We deal with several boundary conditions looking for the appropriate framework to get existence and multiplicity of nontrivial solutions. This kind of equation is related to some models of growth, and for this reason it is natural to study the effect of zero order local reaction terms of the type \(F_{\lambda }(x,u)=\lambda |u|^{p-1}u\), with \(\lambda \in \mathbb {R}\), \(\lambda >0\), and \(0<p<\infty \), and also the solvability of the boundary problems with a source term \(f\) satisfying some integrability hypotheses.

One-Dimensional Stochastic Differential Equations with Generalized Drift

We consider one-dimensional stochastic differential equations with generalized drift which involve the local time $L^X$ of the solution process $ X_t = X_0 + \int_0^t b(X_s) \, {\rm d} B_s + \int_{\bf R} L^X(t,y)$ where $b$ is a measurable real function, $B$ is a Wiener process, and $\nu$ denotes a set function which is defined on the bounded Borel sets of the real line ${\bf R}$ such that it is a finite signed measure on ${\cal B}([-N,N])$ for every $N \in {\bf N}$. This kind of equation is, in dependence of using the right, the left, or the symmetric local time, usually studied under the atom condition $\nu(\{x\}) < 1/2$, $\nu(\{x\}) > -1/2$, and $|\nu(\{x\})| < 1$, respectively. This condition allows us to reduce an equation with generalized drift to an equation without drift and to derive conditions on existence and uniqueness of solutions from corresponding results for equations without drift. The main aim of the present paper is to treat the cases $\nu(\{x\}) \ge 1/2$, $\nu(\{x\}) \le -1/2$, and $|\nu(\{x\})| \ge 1$, respectively, for some $x \in {\bf R}$, and we give a complete description of the features of equations with generalized drift and their solutions in these cases.

Positive Solutions of a Kind of Equations Related to the Laplacian andp-Laplacian

Positive solutions of a kind of equations related to the Laplacian andp-Laplacian on a bounded domain inRNwithN⩾1are studied by using variational method. A sufficient condition of the existence of positive solutions is characterized by the eigenvalues of linear and another nonlinear eigenvalue problems.

О РАЗРЕШИМОСТИ ЗАДАЧИ ПУАНКАРЕ-ТРИКОМИ ДЛЯ УРАВНЕНИЯ СМЕШАННОГО ТИПА ВТОРОГО РОДА

В данной работе впервые доказано однозначная разрешимость нелокальной краевой задачи с условием Пуанкаре для уравнения эллиптико-гиперболического типа второго рода, т.е. для уравнения, где линия вырождения является огибающей семейства характеристик и сама также является характеристикой. Единственность решения задачи доказывается методом интегралов энергии, а существование – методом интегральных уравнений. Библиогр.: 10 назв.

Improve regression-based models for prediction of internal-bond strength of particleboard using Buckingham’s pi-theorem

Internal bond (IB) strength is one of the most important mechanical properties that indicate particleboard quality. The aim of this study was to find a simple regression model that considers the most important parameters that can influence on IB strength. In this study, IB strength was predicted by three kinds of equations (linear, quadratic, and exponential) that were based on the percentage of adhesive (8%, 9.5%, and 11%), particle size (+5, −5 +8, −8 12, and −12 mesh), and density (0.65, 0.7, and 0.75 g/cm3). Our analysis of the results (using SHAZAM 9 software) showed that the exponential function best fitted the experimental data and predicted the IB strength with 18% error. In order decrease the error percentage, the Buckingham Pi theorem was used to build regression models for predicting IB strength based on particle size, density, percentage of adhesive, face-screw withdrawal resistance (SWRf), and edge-screw withdrawal resistance (SWRe). From there, three dimensionless groups were created by Buckingham’s Pi theorem and IB strength was predicted based on multiple regression models. The results showed these models can predict IB strength with 10.68% and 18.17% error, based on face-screw withdrawal resistance and edge-screw withdrawal resistance, respectively.

Fredholm's alternative breaks the confinement of electromagnetic waves

The recovery of information from the so called electromagnetic evanescent waves seems to be a very well explained item. Nevertheless, the travelling waves that becomes from the evanescent waves emerge from integral or differential equations that are very different to those describing the conventional ones. Indeed, we can say that the two kinds of solutions, the travelling and evanescent waves represent a mutually discriminating problem in which we cannot have simultaneous validity of both kinds of equations even they represents the physical evolution of a the same system. But if we can describe our system with a Fredholm's equation we can relate both situations through the properties of the Fredholm's eigenvalue. When the Fredholm's eigenvalue has its values into certain range then Fredholm's equation describes a normal travelling spectrum; otherwise, we are in the presence of another type of equation with abnormal or special behavior. In this work, we analyze the so-named Fredholm's alternative, which enables us to describe the change of positive refraction index-like conditions of broadcasting media to negative refraction index-like conditions. We also sketch some general conditions for the Fredholm's eigenvalue in order to establish general rules for the breaking of the waves’ confinement.

A weighted H1 seminorm regularization method for Fredholm integral equations of the first kind

Many problems in mathematics and engineering lead to Fredholm integral equations of the first kind, e.g. signal and image processing. These kinds of equations are difficult to solve numerically since they are ill-posed. Therefore, regularization is required to obtain a reasonable approximate solution. This paper presents a new regularization method based on a weighted H1 seminorm. Details of numerical implementation are given. Numerical examples, including one-dimensional and two-dimensional integral equations, are presented to illustrate the efficiency of the proposed approach. Numerical results show that the proposed regularization method can restore edges as well as details.

A NEW FORMULA FOR ADOMIAN POLYNOMIALS AND THE ANALYSIS OF ITS TRUNCATED SERIES SOLUTION FOR FRACTIONAL NON-DIFFERENTIABLE INITIAL VALUE PROBLEMS

AbstractA new formula for Adomian polynomials is introduced and applied to obtain truncated series solutions for fractional initial value problems with nondifferentiable functions. These kinds of equations contain a fractional single term which is examined using Jumarie fractional derivatives and fractional Taylor series for nondifferentiable functions. The property of nonlocality of these equations is examined, and the existence and uniqueness of solutions are discussed. Convergence and error analysis for the Adomian series solution are also studied. Numerical examples show the accuracy and efficiency of this formula for solving initial value problems for high-order fractional differential equations.

Symmetries and solutions to geometrical flows

In this paper, we investigate group-invariant solutions to the hyperbolic geometric flow on Riemann surfaces, which include solutions of separation variables, traveling wave solutions, self-similar solutions and radial solutions. In the proceeding of reduction, there are elliptic, hyperbolic and mixed types of equations. For the first kind of equation, some exact solutions are found; while for the last two kinds, with implicit solutions found, we furthermore investigate whether there will be a global solution or blowing up. Referring to the work of Kong et al. (2009), the results come out perfectly.