Problem For Nonlinear Equation Research Articles

Representation of solutions of boundary value problems for nonlinear evolution equations by special series with recurrently calculated coefficients

One of the analytical methods of presenting solutions of nonlinear partial differential equations is the method of special series in powers of specially selected functions called basic functions. The coefficients of such series are found successively as solutions of linear differential equations. The basic functions can also contain arbitrary functions. By using such functional arbitrariness allows in some cases, to prove the global convergence of the corresponding constructed series, and also allows us to prove the solvability of the boundary value problem for the Korteweg-de Vries equation. In the paper for a nonlinear wave equation a theorem on the possibility of satisfying a given boundary condition using an arbitrary function contained in the basic function is proved.

Some regularization methods for a class of nonlinear fractional evolution equations

In this paper, we study initial value problems for nonlinear fractional elliptic equations. In general, the problem is not well-posed, and herein the Hadamard-instability occurs. Under some weak a priori assumptions on the sought solution, we propose two new regularization methods to stabilize the problem when the source term is a globally or locally Lipschitz function. Furthermore, we also investigate the error estimate and show that the approximate solution converges to the exact solution in L2 and H2 norms.

Second-Order Regularity for Parabolic p-Laplace Problems

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

Solvability for a class of nonlinear Hadamard fractional differential equations with parameters

In this paper, we investigate a class of boundary value problem of nonlinear Hadamard fractional differential equations with p-Laplacian operator. By means of the properties of the Green’s functions and Guo–Krasnosel’skii fixed point theorem on cones, various existence results for positive solutions are derived in terms of different values of parameters.

Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions

In the paper the convergence of a finite difference scheme for two-dimensional nonlinear elliptic equation in the rectangular domain with the integral boundary condition is considered. The majorant is constructed for the error of the solution of the system of difference equations, and the estimation of this error is obtained. With this aim, the idea of application of the M-matrices for the theoretical investigation of the system of difference equations was developed. Main results for the convergence of the difference schemes are obtained considering the structure of the spectrum and properties of the M-matrices for a wider class of boundary value problems for nonlinear equations with nonlocal conditions. The main advantage of the suggested method is that the error of approximate solution is estimated in the maximum norm.

Inverse problems for quadratic derivative nonlinear wave equations

For semilinear wave equations on Lorentzian manifolds with quadratic derivative nonlinear terms, we study the inverse problem of determining the background Lorentzian metric. Under some conditions on the nonlinear term, we show that from the source-to-solution map, one can determine the Lorentzian metric up to diffeomorphisms.

Initial-Boundary Value Problems for Generalized Dispersive Equations of Higher Orders Posed on Bounded Intervals

Initial-boundary value problems for nonlinear dispersive equations of evolution of order $$2l+1, \;l\in \mathbb {N}$$ with a convective term of the form $$u^ku_x,\;k\in \mathbb {N}$$ have been considered on intervals $$(0,L),\;L\in (0,+\infty )$$ . Definitions of regular and critical values of k as functions of l have been given. The existence and uniqueness of global regular solutions as well as exponential decay of them for small initial data have been established.

Initial boundary value problem for nonlinear Dirac equation of Gross-Neveu type in 1 + 1 dimensions

This paper studies an initial boundary value problem for a class of nonlinear Dirac equations with cubic terms, which include the equations for the massive Thirring model and the massive Gross-Neveu model. Under the assumptions that the initial data has bounded L2 norm and the boundary satisfies suitable conditions, the global existence and the uniqueness of the strong solution are proved.

Monotone Iterative Technique and Ulam-Hyers Stability Analysis for Nonlinear Fractional Order Differential Equations with Integral Boundary Value Conditions

In this manuscript, the monotone iterative scheme has been extended to the nature of solution to boundary value problem of fractional differential equation that consist integral boundary conditions. In this concern, some sufficient conditions are developed in this manuscript. On the base of sufficient conditions, the monotone iterative scheme combined with lower and upper solution method for the existence, uniqueness, error estimates and various view plots of the extremal solutions to boundary value problem of nonlinear fractional differential equations have been studied. The obtain results have clarified the nature of the extremal solutions. Further, the Ulam--Hyers and Ulam--Hyers--Rassias stability have been investigated for the considered problem. Two illustrative examples of the BVP of the nonlinear fractional differential equations have been provided to justify our contribution.

On the connection problem for nonlinear differential equation

We consider the connection problem of the second nonlinear differential equation \t\t\t1Φ″(x)=(Φ′2(x)−1)cotΦ(x)+1x(1−Φ′(x))\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\varPhi''(x)= \\bigl( \\varPhi^{\\prime2}(x)-1 \\bigr)\\cot\\varPhi(x)+\\frac{1}{x} \\bigl(1- \\varPhi'(x) \\bigr) \\end{aligned}$$ \\end{document} subject to the boundary condition varPhi(x)=x-ax^{2}+O(x^{3}) as xrightarrow0. In view of the fact that equation (1) is equivalent to the fifth Painlevé (PV) equation after a Möbius transformation, we are able to study the connection problem of equation (1) by investigating the corresponding connection problem of PV. Our research technique is based on the method of uniform asymptotics presented by Bassom et al. The asymptotic behavior of the monotonic solution as xrightarrow infty on the real axis of equation (1) is obtained, the explicit relation (connection formula) between the constants appearing in the asymptotic behavior and the real number a are also obtained.

Extremum Principle for the Hadamard Derivatives and Its Application to Nonlinear Fractional Partial Differential Equations

Abstract In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved.

The obstacle problem for nonlinear noncoercive elliptic equations with L^{1}-data

In this paper, we study the obstacle problem governed by nonlinear noncoercive elliptic equations with L^{1}-data. We prove the existence of an entropy solution and show its continuous dependence on the L^{1}-data in W^{1,q}(varOmega ) with q>1.

Existence of Solutions for Boundary Value Problem of Non-linear Integro-differential Equations of Fractional Order

In this article, we cultivate the existence theory for the following boundary value problem of fractional integro-differential equations Dα u(t) = f(t,u(t), (φu)(t)), t ∈ [0,T], 1 <α ≤ 2, (φ u)(t)) = γ(t, s)u(s)ds, together with fractional integro-differential boundary conditions Dα−2u(0+) = 0, Dα−1u(0+) =νIα−1u(η), 0 <η < T. By using the coincidence degree theory, we will obtain a new criteria for the existence of the solutions of the given boundary value problems. We present an example to illustrate our main results.

Harnack Inequalities for Simple Heat Equations on Riemannian Manifolds

In this paper, we consider Harnack inequalities (the gradient estimates) of positive solutions for two different heat equations via the use of the maximum principle. In the first part, we obtain the gradient estimate for positive solutions to the following nonlinear heat equation problem $$\partial _{t}u={\Delta } u+au\log u+Vu,~~u>0$$ on the compact Riemannian manifold (M, g) of dimension n and with Ric(M) ≥−K. Here a > 0 and K are some constants and V is a given smooth positive function on M. Similar results are showed to be true in case when the manifold (M, g) has compact convex boundary or (M, g) is a complete non-compact Riemannian manifold. In the second part, we study Harnack inequality (gradient estimate) for positive solution to the following linear heat equation on a compact Riemannian manifold with non-negative Ricci curvature: $$ \partial _{t}u={\Delta } u+\sum W_{i}u_{i}+Vu, $$ where Wi and V only depend on the space variable x ∈ M. The novelties of our paper are the refined global gradient estimates for the corresponding evolution equations, which are not previously considered by other authors such as Yau (Math. Res. Lett. 2(4), 387–399, 1995), Ma (J. Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010).

Schwarz problem for higher‐order complex partial differential equations in the upper half plane

AbstractLinear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.

The Dirichlet problem for nonlinear elliptic equations with variable exponent

In this paper we study the Dirichlet problem for nonlinear elliptic equations with variable exponents in Sobolev spaces with variable exponent. We show that for every continuous function $ g $ on the boundary there exists a unique continuous extension of $ g $.

Well-posedness of Cauchy problem for Landau equation in critical Besov space

We study the Cauchy problem for the inhomogeneous non linear Landau equation with Maxwellian molecules. In perturbation framework, we establish the global existence of solution in spatially critical Besov spaces. Precisely, if the initial datum is a a small perturbation of the equilibrium distribution in the Chemin-Lerner space $\widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$, then the Cauchy problem of Landau equation admits a global solution belongs to $\widetilde L_t^\infty \widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$. The spectral property of Landau operator enables us to develop new trilinear estimates, which leads to the global energy estimate.

Existence theory for coupled nonlinear third-order ordinary differential equations with nonlocal multi-point anti-periodic type boundary conditions on an arbitrary domain

In this paper, we derive existence and uniqueness results for a coupled system of nonlinear third order ordinary differential equations equipped with nonlocal multi-point anti-periodic type coupled boundary conditions. Leray-Schauder alternative and Banach contraction mapping principle are the main tools of our study. Examples are constructed for illustrating the obtained results. Under appropriate conditions, our results correspond to the ones for an ant-periodic boundary value problem of nonlinear third order ordinary differential equations.

Extremal Solutions for Nonlinear First-Order Impulsive Integro-Differential Dynamic Equations

This paper is concerned with the initial-value problem for nonlinear first-order impulsive integro-differential equations on time scales $$\mathbb{T}$$ . We establish certain existence criteria by using a fixed-point theorem for operator on cones, under which such problems have aminimal and amaximal solution lying in a corresponding region bounded by their lower and upper solutions.

HP estimation for the Cauchy problem for nonlinear elliptic equation

In this paper, we investigate the Cauchy problem for a ND nonlinear elliptic equation in a bounded domain. As we know, the problem is severely ill-posed. We apply the Fourier truncation method to regularize the problem. Error estimates between the regularized solution and the exact solution are established in Hp space under some priori assumptions on the exact solution.