Problem For Nonlinear Equation Research Articles

Local Solvability, Blow-up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: I. Green Formulas

Local Solvability, Blow-up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: I. Green Formulas

On the blowup of solutions of the Cauchy problem for nonlinear equations of ferroelectricity theory

On the blowup of solutions of the Cauchy problem for nonlinear equations of ferroelectricity theory

Existence of Solutions of Boundary Value Problem for Nonlinear One-Dimensional Wave Equations by Fixed Point Method

In this article, we investigate the solutions for a class of initial boundary value problems for nonlinear one-dimensional wave equations using a fixed point method. The findings consider initial and boundary data which lead to determining the nature of classes of the nonlinear one-dimensional wave equation. The obtained results indicate the availability of nonnegative solutions which are proved using the method of the fixed point. A new fixed point approach is effective and used as a definitive modeling method to prove the existence of nonnegative solutions for a class of initial boundary value problems based on recent theoretical arguments. The results in this study are considered with examples.

The Taylor series method of order p and Adams–Bashforth method on time scales

A recent study on the Taylor series method of second order and the trapezoidal rule for dynamic equations on time scales has been continued by introducing a derivation of the Taylor series method of arbitrary order on time scales. The error and convergence analysis of the method is also obtained. The 2‐step Adams–Bashforth method for dynamic equations on time scales is concluded and applied to examples of initial value problems for nonlinear dynamic equations. Numerical results are presented and discussed.

On boundary value problems for nonlinear elliptic equations on unbounded domains

On boundary value problems for nonlinear elliptic equations on unbounded domains

On a System Without Critical Points Arising in Heat Conductivity Theory

A two-point boundary value problem for the second order nonlinear ordinary differential equation, arising in the heat conductivity theory, is considered. Multiplicity and existence results are established for this problem, where the equation contains two parameters.

NUMERICAL METHOD FOR SOLVING THE DIRICHLET BOUNDARY VALUE PROBLEM FOR NONLINEAR TRIHARMONIC EQUATION

In this work, we consider the Dirichlet boundary value problem for nonlinear triharmonic equation. Due to the reduction of the problem to operator equation for the pair of the right hand side function and the unknown second normal derivative of the function to be sought, we design an iterative method at both continuous and discrete levels for numerical solution of the problem. Some examples demonstrate that the numerical method is of fourth order convergence. When the right hand side function does not depend on the unknown function and its derivatives, the numerical method gives more accurate results in comparison with the results obtained by the interior method of Gudi and Neilan.

Nonlinearizable solutions in an eigenvalue problem for Maxwell's equations with nonhomogeneous nonlinear permittivity in a layer

AbstractThe paper focuses on a problem arising in nonlinear guided wave optics. The problem for Maxwell's equations with nonhomogeneous nonlinear constitutive law is reduced to an eigenvalue problem for a system of nonlinear ordinary differential equations with local boundary conditions. We suggest a novel approach to study eigenvalue problems for nonlinear nonautonomous equations based on studying an integral characteristic equation (ICE) of the problem. Using asymptotical analysis of the ICE, we prove existence of infinitely many eigenvalues and find their distribution. It is important that the corresponding linear problem has only a finite number of eigenvalues.

On the Boundary Value Problem of Nonlinear Fractional Integro-Differential Equations

Using Banach’s contractive principle and the Laray–Schauder fixed point theorem, we study the uniqueness and existence of solutions to a nonlinear two-term fractional integro-differential equation with the boundary condition based on Babenko’s approach and the Mittag–Leffler function. The current work also corrects major errors in the published paper dealing with a one-term differential equation. Furthermore, we provide examples to illustrate the application of our main theorems.

The Obstacle Problem For Nonlinear Degenerate Elliptic Equations with Variable Exponents and L$^1$-Data

The Obstacle Problem For Nonlinear Degenerate Elliptic Equations with Variable Exponents and L$^1$-Data

AN INITIAL-BOUNDARY VALUE PROBLEM FOR KELVIN-VOIGT EQUATIONS WITH (p(x), q(x),m(x))- STRUCTURE

The proof of the global in time existence of solutions of initial-boundary value problems for nonlinear equations in many The proof of the global in time existence of solutions of initial-boundary value problems for nonlinear equations in many cases,  is not easy, even in some cases it is impossible. However, by showing some qualitative properties of its solution, one can find answers to such questions. For example, by establishing the blow up in a finite time property of a solution, one can show that a solution does not exist globally in time. Such way, in last years, the investigating the quality properties of solutions as localization and/or blow up in a finite time, has been developing rapidly In this paper, we study the initial-boundary value problem for the Kelvin-Voigt equations with both of diffusion and relaxation term, modified by (p(x),q(x))-Laplacian, respectively, and with the variable exponent damping term. The damping term in the equation realizes as nonlinear source term. In this work, we study the nonlinear initial-boundary value problem for generalized Kelvin-Voigt equations describing the motion of incompressible viscoelastic non-Newtonian fluids. The equations generalized by replacing the diffusion and relaxation terms in equation with p(x)-Laplacian and q(x)-Laplacian, respectively, and adding a nonlinear absorption term with variable exponents and coefficients. The definition of a weak solution is given. Under suitable conditions for variable exponents and coefficients, and data of the problem, the blow up of weak solutions are established.

Rigorous numerics for nonlinear heat equations in the complex plane of time

In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

On the Existence of Solutions to Boundary Value Problems for Nonlinear Equilibrium Equations of Shallow Anisotropic Shells of Timoshenko Type in Sobolev Space

On the Existence of Solutions to Boundary Value Problems for Nonlinear Equilibrium Equations of Shallow Anisotropic Shells of Timoshenko Type in Sobolev Space

Differentiation Functors and Category Interpretation of Optimality Conditions

Operator derivatives are determined as functors. Necessary optimality conditions with category interpretation are proved for abstract optimization control problems. Finite dimensional extremum problems and an optimization control problem for nonlinear parabolic equation with state constraints are considered as examples.

Mixed boundary valued problems for linear and nonlinear wave equations in domains with fractal boundaries

The weak well-posedness, with the mixed boundary conditions, of the strongly damped linear wave equation and of the non linear Westervelt equation is proved in a large natural class of Sobolev admissible non-smooth domains. In the framework of uniform domains in \(\mathbb {R}^2\) or \(\mathbb {R}^3\) we also validate the approximation of the solution of the Westervelt equation on a fractal domain by the solutions on the prefractals using the Mosco convergence of the corresponding variational forms.

Long time existence for two-dimension elastic waves

In this paper, for the Cauchy problem of nonlinear elastic wave equations for two-dimension isotropic, homogeneous and hyperelastic materials with small initial data, we give the lifespan estimates for classical solutions.

Dirichlet boundary valued problems for linear and nonlinear wave equations on arbitrary and fractal domains

We obtain the weak well-posedness results for the linear strongly damped wave equation and the nonlinear Westervelt equation on arbitrary three-dimensional domains with homogeneous Dirichlet boundary conditions. In R2, we prove the well-posedness in the class of NTA domains or their limit domains, obtained as a limit of sequences of NTA domains, characterized by the same geometrical constants. The nonhomogeneous Dirichlet condition is also treated for Sobolev extension domains of Rn with a d-set boundary n−2<d<n preserving Markov's local inequality. For a converging sequence of domains in the sense of characteristic functions, we establish the Mosco convergence of the functionals corresponding to the weak formulations for the Westervelt equation with the homogeneous Dirichlet boundary condition.

Carleman estimates and the contraction principle for an inverse source problem for nonlinear hyperbolic equations

The main aim of this paper is to solve an inverse source problem for a general nonlinear hyperbolic equation. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the inverse problem. To find this fixed point, we define a recursive sequence with an arbitrary initial term by the same manner as in the classical proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution with the exponential rate. Therefore, our new method can be considered as an analog of the contraction principle. We rigorously study the stability of our method with respect to noise. Numerical examples are presented.

Terminal value problem for nonlinear parabolic equation with Gaussian white noise

&lt;abstract&gt;&lt;p&gt;In this paper, We are interested in studying the backward in time problem for nonlinear parabolic equation with time and space independent coefficients. The main purpose of this paper is to study the problem of determining the initial condition of nonlinear parabolic equations from noisy observations of the final condition. The final data are noisy by the process involving Gaussian white noise. We introduce a regularized method to establish an approximate solution. We establish an upper bound on the rate of convergence of the mean integrated squared error. This article is inspired by the article by Tuan and Nane &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b1"&gt;1&lt;/xref&gt;]&lt;/sup&gt;.&lt;/p&gt;&lt;/abstract&gt;

ДВОСТОРОННI МЕТОДИ РОЗВ’ЯЗУВАННЯ ПОЧАТКОВОЇ ЗАДАЧI ДЛЯ НЕЛIНIЙНИХ IНТЕГРО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ

Using the continued fractions and the method of constructing Runge-Kutta methods, numerical methods for solving the Cauchy problem for nonlinear Volterra non-linear integrodifferential equations are proposed. With appropriate values of the parameters, one can obtain an approximation to the exact solution of the first and second order of accuracy. We found a set of parameters for which we obtain two-sided calculation formulas, which at each step of integration allow to obtain the upper and lower approximations of the exact solution.