Problem For Nonlinear Equation Research Articles

About the Approximate Solutions to Linear and Non-Linear Pseudodifferential Reaction Diffusion Equations

Background. The concept of fractal is one of the main paradigms of modern theoretical and experimental physics, radiophysics and radar, and fractional calculus is the mathematical basis of fractal physics, geothermal energy and space electrodynamics. We investigate the solvability of the Cauchy problem for linear and nonlinear inhomogeneous pseudodifferential diffusion equations. The equation contains a fractional derivative of a Riemann–Liouville time variable defined by Caputo and a pseudodifferential operator that acts on spatial variables and is constructed in a homogeneous, non-negative homogeneous order, a non-smooth character at the origin, smooth enough outside. The heterogeneity of the equation depends on the temporal and spatial variables and permits the Laplace transform of the temporal variable. The initial condition contains a restricted function. Objective. To show that the homotopy perturbation transform method (HPTM) is easily applied to linear and nonlinear inhomogeneous pseudodifferential diffusion equations. To prove the solvability and obtain the solution formula for the Cauchy problem series for the given linear and nonlinear diffusion equations. Methods. The problem is solved by the NPTM method, which combines a Laplace transform with a time variable and a homotopy perturbation method (HPM). After the Laplace transform, we obtain an integral equation which is solved as a series by degrees of the entered parameter with unknown coefficients. Substituting the input formula for the solution into the integral equation, we equate the expressions to equal parameter degrees and obtain formulas for unknown coefficients. When solving the nonlinear equation, we use a special polynomial which is included in the decomposition coefficients of the nonlinear function and allows the homotopy perturbation method to be applied as well for nonlinear non-uniform pseudodifferential diffusion equation. Results. The result is a solution of the Cauchy problem for the investigated diffusion equation, which is represented as a series of terms whose functions are found from the parametric series. Conclusions. In this paper we first prove the solvability and obtain the formula for solving the Cauchy problem as a series for linear and nonlinear inhomogeneous pseudodifferential equations.

Evolution equations and diffusion operators for demographic dynamics

ABSTRACT In this paper, we first review the related results of the evolution equations and operators in the parity theory of demographic evolution, and respectively establish differential equations and difference equations based on different initial states. In order to consider regional constraints, we modify the differential equation model to a stochastic Yule model and analyze the main features of the solution. Then, we add nonlinear terms to represent the limiting factors, transform the differential equation model into the initial value problem of nonlinear ordinary differential equations with parameters, and show the existence, stability and asymptotic behavior of global classical solutions in () spaces. Finally, we further study the demographic operator and the corresponding heat equation, and give the explicit structure of the singular geodesic of the associated Hamiltonian system and singular heat kernel of the operator.

Summability of solutions of the Dirichlet problem for nonlinear elliptic equations with right-hand side in logarithmic classes close to [formula omitted

In this paper, we consider the Dirichlet problem for a class of nonlinear elliptic second-order equations with right-hand side in L1. We describe the summability properties of entropy and weak solutions of this problem in the case where the right-hand side belongs to classes close to L1 and defined by the logarithmic function or its compositions of arbitrary multiplicity.

A Study on Impulsive Hilfer Fractional Evolution Equations with Nonlocal Conditions

Abstract In this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and more general to known results. At last, an example is provided to illustrate the results.

Local Solution of Delay Fractional Differential Equations

By Schauder’s Fixed Point Theorem and some new methods, we study the existence and uniqueness of initial value problems for nonlinear higher fractional equations with delay, and obtain some new results on local solutions.

The Ablowitz-Ladik system on a graph

This paper presents an approach to study initial-boundary value (IBV) problems for integrable nonlinear differential-difference equations (DDEs) posed on a graph. As an illustrative example, we consider the Ablowitz–Ladik system posed on a graph that is constituted by N semi-infinite lattices (edges) connected through some boundary conditions. We first show that analyzing this problem is equivalent to analyzing a certain matrix IBV problem; then we employ the unified transform method (UTM) to analyze this matrix IBV problem. We also compare our results with some previously known studies. In particular, we show that the inverse scattering method (ISM) for the integrable DDEs on the set of integers can be recovered from the UTM applied to our N = 2 graph problem as a particular case, and the non-local reductions of integrable DDEs can be obtained as local reductions from our results.

Estimates for solutions to nonlinear degenerate elliptic equations with lower order terms

We consider a class of Dirichlet boundary problems for nonlinear degenerate elliptic equations with lower order terms. We prove, using symmetrization techniques, pointwise estimates for the rearrangement of the gradient of a solution u and integral estimates. As consequence, we get apriori estimates which show how the summability of the gradient of a solution increases when the summability of the datum increases, also taking into account the presence of a zero order term which can have a regularizing effect.

Nonlinear Evolutionary Schrödinger Equation in a Two-Dimensional Domain

We consider a mixed problem for a nonlinear evolutionary Schrodinger equation in a two-dimensional domain and study the smoothness of solutions and their destruction.

On Weak Solution of Cauchy Problem for Nonlinear Parabolic Equations

In this paper, we investigate weak solutions of Cauchy problem for nonlinear parabolic equations. By using Galerkin approximation, the existence and uniqueness of weak solutions are proved.

Linear and nonlinear degenerate differential operators and applications

The boundary value and mixed value problems for linear and nonlinear singular degenerate abstract elliptic and parabolic equations are studied. The linear problems involve some parameters. The uniform Lp- separability properties of linear problems and the optimal regularity results for nonlinear problems are obtained. The equations include linear operators defined in Banach spaces, in which by choosing the spaces and operators we can obtain numerous classes of problems for singular degenerate differential equations which occur in a wide variety of physical systems.

A modified BFGS type quasi-Newton method with line search for symmetric nonlinear equations problems

In this paper, using approximate gradient of the norm square metric function, we present an inexact MBFGS method with line search for solving symmetric nonlinear equations, which is a generalization of the MBFGS method proposed by Li and Fukushima (2001) for solving smooth unconstrained optimization. The used approximate gradient of the proposed method only requires computing two residuals at every iteration and is easy to obtain by utilizing symmetric structure of the system. We establish global convergence of the proposed method and report some numerical results.

Initial boundary value problems for some nonlinear dispersive models on the half-line: a review and open problems

In the last years the study of initial boundary value problems for nonlinear dispersive equations on the half-lines has given attention of many researchers. This turns out to be a rather challenging problem, mainly when studied in low Sobolev regularity. In this note we present a review of the main results about this topic and also introduce interesting open problems which still requires attention from the mathematical point of view.

EXISTENCE AND STABILITY RESULTS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO FRACTIONAL DERIVATIVES

In this paper, we discuss the existence, uniqueness and stability of solutions for a nonlocal boundary value problem of nonlinear fractional differential equations with two Caputo fractional derivatives. By applying the contraction mapping and O’Regan fixed point theorem, the existence results are obtained. We also derive the Ulam-Hyers stability of solutions. Finally, some examples are given to illustrate our results.

On Cauchy problem for nonlinear fractional differential equation with random discrete data

This paper is concerned with finding the solution u(x, t) of the Cauchy problem for nonlinear fractional elliptic equation with perturbed input data. This study shows that our forward problem is severely ill-posed in sense of Hadamard. For this ill-posed problem, the trigonometric of non-parametric regression associated with the truncation method is applied to construct a regularized solution. Under prior assumptions for the exact solution, the convergence rate is obtained in both L2 and Hq (for q > 0) norm. Moreover, the numerical example is also investigated to justify our results.

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.