Problem For Nonlinear Equation Research Articles

φ − (h,e)-concave operators and applications

In this article, by introducing a new set and a new concept of φ−(h,e)-concave operators, and by using the cone theory and monotone iterative method, we present some new existence and uniqueness theorems of fixed points for increasing φ−(h,e)-concave operators without requiring the existence of upper and lower solutions. As an application, we establish the existence and uniqueness of a nontrivial solution for a new form of fractional differential equation with integral boundary conditions. The main results of this paper improve and extend some known results, and present a new method to study nonlinear equation problems.

Monte Carlo method for solution of initial–boundary value problem for nonlinear parabolic equations

In this paper we will consider the initial–boundary value problem for a parabolic equation with a polynomial non-linearity relative to the unknown function. First we will derive a probabilistic representation of our problem. The representation of the solution of this problem is given in the form of a mathematical expectation, which is determined based on trajectories of branching processes. Under the assumption of the existence of the solution, an unbiased estimator is built using trajectories of a branching process. We will use a mean value theorem to write out a special integral equation, that equates the value of the unknown function u(x,t) with its integral over a spheroid and balloid with center at the point (x,t). A probabilistic representation of the solution to the problem in the form of mathematical expectation of some random variables is obtained. This probabilistic representation uses a branching process whose trajectories are used in the contraction of an unbiased estimator for the solution. The derived unbiased estimator has a finite variance, and is built up from trajectories of branching processes with a finite average number of branches. Finally, the results of numerical experiments and application to the practical problems are discussed.

Ulam Stability for Fractional Partial Integro-Differential Equation with Uncertainty

In this paper, the solvability of Darboux problems for nonlinear fractional partial integro-differential equations with uncertainty under Caputo gH-fractional differentiability is studied in the infinity domain J∞ = [0,∞) × [0,∞). New concepts of Hyers-Ulam stability and Hyers-Ulam-Rassias stability for these problems are also investigated through the equivalent integral forms. A computational example is presented to demonstrate our main results.

A boundary value problem for nonlinear differential equation with arbitrary functions

A boundary value problem for nonlinear differential equation with arbitrary functions

Identification of the initial condition in backward problem with nonlinear diffusion and reaction

In this paper, we study the backward in time problem for nonlinear parabolic equations associated with nonlinear diffusion coefficient. The problem is ill-posed in the sense of Hadamard. Under an a priori assumption on the exact solution in Gevrey space, we propose a new regularization method for stabilising the ill-posed problem. Our results extend some earlier works on backward problems for nonlinear parabolic equations.

Some Theorems on Cauchy Problem for Nonlinear Fractional Differential Equations with Positive Constant Coefficient

In this paper, we consider a Cauchy problem for nonlinear fractional differential equation with constant coefficient \(\lambda >0\) of the type: \({^c}{D}^{\alpha }x(t)=\lambda x(t)+f(t,x(t))\) with \(x(0)=x_{0}.\) The aim of this paper is to investigate the existence and interval of existence of solutions, uniqueness, continuous dependence of solutions on initial conditions, estimates on solutions and continuous dependence on parameters and functions involved in the equations. Finally, one illustrative example is given to demonstrate the theoretical results.

The Inverse Problem for the Nonlinear Pseudoparabolic Equation of Filtration Type

The Inverse Problem for the Nonlinear Pseudoparabolic Equation of Filtration Type

Stability of the Size-Dependent and Functionally Graded Curvilinear Timoshenko Beams

The size-dependent model is studied based on the modified couple stress theory for the geometrically nonlinear curvilinear Timoshenko beam made from a functionally graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the curvilinear beams is investigated with the use of the setup method. The second-order accuracy finite difference method is used to solve the problem of nonlinear partial differential equations (PDEs) by reducing it to the Cauchy problem. The obtained set of nonlinear ordinary differential equations (ODEs) is then solved by the fourth-order Runge–Kutta method. The relaxation method is employed to solve numerous static problems based on the dynamic approach. Eight different combinations of size-dependent coefficients and the functionally graded material coefficient are used to study the stress-strain responses of Timoshenko beams. Stability loss of the curvilinear Timoshenko beams is investigated using the Lyapunov criterion based on the estimation of the Lyapunov exponents. Beams with/without the size-dependent behavior, homogeneous beams, and functionally graded beams having the same stiffness are investigated. It is shown that in straight-line beams, the size-dependent effect decreases the beam deflection. The same is observed if the most rigid layer is located on the top of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid layer essentially improves the beam strength against stability loss. The observed transition of the largest Lyapunov exponent from a negative to positive value corresponds to the transition from a precritical to postcritical beam state.

Boundary-Value Problems for Nonlinear Parabolic Equations with Delay and Degeneration at the Initial Time

We study boundary-value problems with Dirichlet conditions for nonlinear parabolic equations with variable delay (i.e., delay is a function of time) and degeneration at the initial time. The existence and uniqueness of the classical solution of this problem are proved. The a priori estimates of this solution are obtained.

A note on the derivation of filter regularization operators for nonlinear evolution equations

Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous studies cannot be clearly adapted to handle more complex scenarios, albeit the greatly increasing concerns on the improvement of wider classes. In this note, we rigorously study a general theory for filter regularized operators in a Hilbert space for nonlinear evolution equations which have occurred naturally in different areas of science. The starting point lies in problems that are in principle ill-posed with respect to the initial/final data – these basically include the Cauchy problem for nonlinear elliptic equations and the backward-in-time nonlinear parabolic equations. We derive general filters that can be used to stabilize those problems. Essentially, we establish the corresponding well-posed problem whose solution converges to the solution of the ill-posed problem. The approximation can be confirmed by the error estimates in the Hilbert space. This work improves very much many papers in the same field of research.

Positive solutions for eigenvalue problem of non-linear fractional differential equations

In this paper, by using the fixed point theorem of cone expansion and compression of norm type and monotone iterative technique, we study the eigenvalue problem of non-linear fractional differential equations. The existence and iteration of positive solutions are obtained.

非线性方程组问题的一个三项共轭梯度算法

本文设计一个三项共轭梯度算法，用来求解非线性方程组问题，建议的算法优点是：(1) 搜索方向的充分下降性；(2) 搜索方向的信赖域特点；(3) 算法能保证具有全局收敛性；(4) 算法能成功求解一千维以上的非线性方程组问题。 This paper designs a three-term conjugate gradient algorithm for nonlinear equations problems and the proposed algorithm has four advantages: (1) the sufficient descent property is satisfied for the search direction; (2) the trust region feature holds for the direction too; (3) the global convergence of the proposed algorithm is possessed; (4) the new algorithm can successfully solve nonlinear equations problems with 1000 dimensions.

On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation

In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, by “Concavity” method we establish three blow-up results for certain solutions in the case 1): , in the case 2): and in the case 3): . At last, we consider that the estimation of the upper bounds of the blow-up time is given for deferent initial energy.

Existence of Positive Solutions to a Singular Semipositone Boundary Value Problem of Nonlinear Fractional Differential Systems

In this paper, we consider the existence of positive solutions to a singular semipositone boundary value problem of nonlinear fractional differential equations. By applying the fixed point index theorem, some new results for the existence of positive solutions are obtained. In addition, an example is presented to demonstrate the application of our main results.

Existence results of nonlocal boundary value problems for nonlinear fractional q-integrodifference equations

Existence results of nonlocal boundary value problems for nonlinear fractional q-integrodifference equations

Positive solutions for eigenvalue problem of non-linear fractional differential equations

Positive solutions for eigenvalue problem of non-linear fractional differential equations

On approximate solutions to the Neumann elliptic boundary value problem with non-linearity of exponential type

We discuss the existence of weak solutions to one class of Neumann boundary value problems (BVP) for non-linear elliptic equations. We introduce a special family of perturbed optimal control problems (OCPs) where the class of fictitious controls is closely related with the properties of the distribution in the right-hand side of the elliptic equation, and we show that optimal solutions of such problems allow one to attain (in the limit) some approximate solutions as the parameter of perturbation $\varepsilon>0$ tends to zero. The main questions we discuss in this paper touch on the solvability of perturbed OCPs, the uniqueness of their solutions, the asymptotic properties of optimal pairs as the perturbation parameter $\varepsilon >0$ tends to zero, and deriving of optimality conditions for the perturbed OCPs. As a consequence, we obtain the sufficient conditions of the existence of weak solutions to the given class of non-linear Neumann BVPs and propose a way for their approximation.

A general class of periodic boundary value problems for controlled nonlinear impulsive evolution equations on Banach spaces

This paper deals with the periodic boundary value problems for nonlinear impulsive evolution equations with controls. By using the theory of semigroup and fixed point methods, we address some conditions ensuring the existence and uniqueness. Finally, two examples are provided to prove the effectiveness of the proposed results.

On a three point boundary value problem of arbitrary order

In this paper, we study a three point boundary value problem of nonlinear fractional differential equations of order α, 2 < α < 3. New existence and uniqueness results are established using Banach fixed point theorem. Other existence results are obtained using Schaefer and Krasnoselskii theorems. Some illustrative examples are also presented.

Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations

We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in $\mathbb{R}^N$ with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For $L^2$ initial data that are radial and non-increasing as a function of the distance to the origin, we characterize the ignition behavior in terms of the long time behavior of the energy associated with the solution. We then use this characterization to establish existence of a sharp threshold for monotone families of initial data in the considered class under various assumptions on the nonlinearities and spatial dimension. We also prove that for more general initial data that are sufficiently localized the solutions that exhibit ignition behavior propagate in all directions with the asymptotic speed equal to that of the unique one-dimensional variational traveling wave.