Problem For Nonlinear Equation Research Articles

GEKKO Optimization Suite

This paper introduces GEKKO as an optimization suite for Python. GEKKO specializes in dynamic optimization problems for mixed-integer, nonlinear, and differential algebraic equations (DAE) problems. By blending the approaches of typical algebraic modeling languages (AML) and optimal control packages, GEKKO greatly facilitates the development and application of tools such as nonlinear model predicative control (NMPC), real-time optimization (RTO), moving horizon estimation (MHE), and dynamic simulation. GEKKO is an object-oriented Python library that offers model construction, analysis tools, and visualization of simulation and optimization. In a single package, GEKKO provides model reduction, an object-oriented library for data reconciliation/model predictive control, and integrated problem construction/solution/visualization. This paper introduces the GEKKO Optimization Suite, presents GEKKO’s approach and unique place among AMLs and optimal control packages, and cites several examples of problems that are enabled by the GEKKO library.

Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

Multiplicity Results of Multi-Point Boundary Value Problem of Nonlinear Fractional Differential Equations

Multiplicity Results of Multi-Point Boundary Value Problem of Nonlinear Fractional Differential Equations

The Cauchy Problem for Non-linear Higher Order Hartree Type Equation in Modulation Spaces

We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity $$F(u) = (K \star |u|^{2k})u$$ under a specified condition on potential K with Cauchy data in modulation spaces $$M^{p,q}(\mathbb {R}^n)$$ . We establish global well-posedness results in $$M^{1,1}(\mathbb {R}^n)$$ , when $$K(x)= \frac{\lambda }{|x|^{\nu }} ~ (\lambda \in \mathbb {R}, ~ 0< \nu < min\{2, \frac{n}{2}\})$$ , for $$k < \frac{n+2-\nu }{n};$$ and local well-posedness results in $$M^{1,1}(\mathbb {R}^n)$$ , when $$K(x)= \frac{\lambda }{|x|^{\nu }} ~ (\lambda \in \mathbb {R}, ~ 0< \nu < n),$$ for $$k \in \mathbb {N};$$ in $$M^{p,q}(\mathbb {R}^n)$$ with $$1 \le p\le 4,~ 1\le q \le \frac{2^{2k-2}}{2^{2k-2}-1},~k \in \mathbb {N}$$ , when $$K \in M^{\infty , 1}(\mathbb {R}^n)$$ . Moreover, we also consider the Cauchy problem for the non-linear higher order Hartree equations on modulation spaces $$M^{p,1}(\mathbb {R}^n),$$ when $$K \in M^{1, \infty }(\mathbb {R}^n)$$ and show the existence of a unique global solution by using integrability of time decay factors of Strichartz estimates. As a consequence, we are able to deal with wider classes of a nonlinearity and a solution space.

OPTIMAL CONTROL OF NONLINEAR DUFFING OSCILLATORS: LIE-GROUP APPROACHES

In the optimal control theory, the Hamiltonian formulation is a famous one convenient to find an optimal designed control force. However, when the performance index is a complicated function of control force, the Hamiltonian method is not easy to find the optimal solution, because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations (DAEs). In this paper we address this issue via a quite novel and effective approach, of which the optimally controlled vibration problem of Duffing oscillator is recast into a two-point nonlinear DAEs by identifying the unknown control force. We develop the corresponding SL(n, R) and GL(n, R) shooting methods, as well as a Lie-group differential algebraic equations (LGDAE) method to numerically solve the optimal control forces. Eight examples of a single Duffing oscillator and one coupled Duffing oscillators are used to test the performance of the present method.

The Dirichlet problem for the [formula omitted]-fractional Laplace equation

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s∈(0,1) and summability growth p∈(1,∞), whose model is the fractional p-Laplacian operator with measurable coefficients. We review several recent results for the corresponding weak solutions/supersolutions, as comparison principles, a priori bounds, lower semicontinuity, boundedness, Hölder continuity up to the boundary, and many others. We then discuss the good definition of (s,p)-superharmonic functions, and the nonlocal counterpart of the Perron method in nonlinear Potential Theory, together with various related results. We briefly mention some basic results for the obstacle problem for nonlinear integro-differential equations. Finally, we present the connection amongst the fractional viscosity solutions, the weak solutions and the aforementioned (s,p)-superharmonic functions, together with other important results for this class of equations when involving general measure data, and a surprising fractional version of the Gehring lemma.We sketch the corresponding proofs of some of the results presented here, by especially underlining the development of new fractional localization techniques and other recent tools. Various open problems are listed throughout the paper.

Comparison of the results of the modeling of the temperature field in the process of bimetal melting by the scanning laser for the rays, which focused in the rectangle and in the circle

The problem of numerical simulation of temperature field in the process of bimetal melting by the scanning laser for rays focusing in a rectangle and in a circle are considered. The mathematical model of the process is a boundary problem for unsteady three-dimensional nonlinear partial differential equation in the variable domain. Comparison of results of the modeling with different laser beam shape and power distribution is made according to key characteristics

Blow-up problems for nonlinear parabolic equations on locally finite graphs

Let G=(V,E) be a locally finite connected weighted graph, and Δ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut=Δu + f(u) on G. The blow-up phenomenons for ut=Δu + f(u) are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.

Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary

We will study a free boundary problem of the nonlinear diffusion equations of the form ut=uxx+f(u),t>0,ct<x<h(t), where f is C1 function satisfying f(0)=0, c>0 is a given constant and h(t) is a free boundary which is determined by a Stefan-like condition. This model may be used to describe the spreading of a new or invasive species with population density u(t,x) over a one dimensional habitat. The free boundary x=h(t) represents the spreading front. In this model, we impose zero Dirichlet boundary condition at left moving boundary x=ct. This means that the left boundary of the habitat is a very hostile environment for the species and that the habitat is eroded away by the left moving boundary at constant speed c.In this paper we will extend the results of a trichotomy result obtained in [23] to general monostable, bistable and combustion types of nonlinearities. We show that the long-time dynamical behavior of solutions can be expressed by unified fashion, that is, for any initial data, the unique solution exhibits exactly one of the behaviors, spreading, vanishing and transition. We also give the asymptotic profile of the solution over the whole domain when spreading happens. The approach here is quite different from that used in [23].

Positive solutions for a class of nonlinear Hadamard fractional differential equations with a parameter

In this paper, we investigate a class of boundary value problem of nonlinear Hadamard fractional differential equations with a parameter. By means of the properties of the Green function and Guo–Krasnosel’skii fixed-point theorem on cones, the existence and nonexistence of positive solutions are obtained. Finally, some examples are presented to show the effectiveness of our main results.

Gravitational Field as a Pressure Force from Logarithmic Lagrangians and Non-Standard Hamiltonians: The Case of Stellar Halo of Milky Way

Recently, the notion of non-standard Lagrangians was discussed widely in literature in an attempt to explore the inverse variational problem of nonlinear differential equations. Different forms of non-standard Lagrangians were introduced in literature and have revealed nice mathematical and physical properties. One interesting form related to the inverse variational problem is the logarithmic Lagrangian, which has a number of motivating features related to the Liénard-type and Emden nonlinear differential equations. Such types of Lagrangians lead to nonlinear dynamics based on non-standard Hamiltonians. In this communication, we show that some new dynamical properties are obtained in stellar dynamics if standard Lagrangians are replaced by Logarithmic Lagrangians and their corresponding non-standard Hamiltonians. One interesting consequence concerns the emergence of an extra pressure term, which is related to the gravitational field suggesting that gravitation may act as a pressure in a strong gravitational field. The case of the stellar halo of the Milky Way is considered.

Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations

We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are:

Approximate Solutions of Inverse Problems for Nonlinear Space Fractional Diffusion Equations with Randomly Perturbed Data

This paper is concerned with the backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the unstable solution, we de...

Non-existence of global solutions to nonlinear wave equations with positive initial energy

We consider the Cauchy problem for nonlinear abstract wave equations in a Hilbert space. Our main goal is to show that this problem has solutions with arbitrary positive initial energy that blow up in a finite time. The main theorem is proved by employing a result on growth of solutions of abstract nonlinear wave equation and the concavity method. A number of examples of nonlinear wave equations are given. A result on blow up of solutions with arbitrary positive initial energy to the initial boundary value problem for the wave equation under nonlinear boundary conditions is also obtained.

Mixed problems for degenerate abstract parabolic equations and applications

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the unified transform method

We study an initial value problem for the one-dimensional non-stationary linear Schr\"odinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem. Since the unified approach is known as providing efficient solutions to both linear and nonlinear problems, the present paper can be viewed as a step in solving the initial value problem for the non-stationary {\em nonlinear} Schr\"odinger equation with a point singular potential.

A new class of four-point fractional sum boundary value problems for nonlinear sequential fractional difference equations involving shift operators

In this article, we study the existence result for a Riemann-Liouville fractional difference equation with four-point fractional sum boundary value conditions, by using the Sadovskii's fixed point theorem. Our problem contains the shift operators on fractional difference operators that are different orders. Finally, we present an example to show the importance of these result.

Global Nonexistence for the Cauchy Problem of Nonlinear Fractional Hyperbolic Equations and Systems

Global Nonexistence for the Cauchy Problem of Nonlinear Fractional Hyperbolic Equations and Systems

On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain $Ω$ . We take the coefficient $u(x)∈ L^∞(Ω)\cap BV(Ω)$ in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix $A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$ . We show that, in spite of unboundedness of the non-linear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices $A^{skew}$ . We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.

Comparison of Some Iterative Methods of Solving Nonlinear Equations

This work focuses on nonlinear equation (x) = 0, it is noted that no or little attention is given to nonlinear equations. The purpose of this work is to determine the best method of solving nonlinear equations. The work outlined four methods of solving nonlinear equations. Unlike linear equations, most nonlinear equations cannot be solved in finite number of steps. Iterative methods are being used to solve nonlinear equations. The cost of solving nonlinear equations problems depend on both the cost per iteration and the number of iterations required. Derivations of each of the methods were obtained. An example was illustrated to show the results of all the four methods and the results were collected, tabulated and analyzed in terms of their errors and convergence respectively. The results were also presented in form of graphs. The implication is that the higher the rate of convergence determines how fast it will get to the approximate root or solution of the equation. Thus, it was recommended that the Newton’s method is the best method of solving the nonlinear equation f(x) = 0 containing one variable because of its high rate of convergence.