Solutions Of Parabolic Equations Research Articles

Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0<ɛ<<1) is multiplied the highest order derivative term. The problem’s solution shows an exponential boundary layer on the right side of the spatial domain for small values of ɛ. The solution and its derivatives are discussed together with their properties and bounds. The Crank–Nicolson approach for time discretization and the fitted operator non-standard finite difference method for space discretization are used to solve the time fractional singularly perturbed time delay problem under consideration. The comparison principle and solution bound are used to examine and analyze the scheme’s stability. The scheme’s uniform convergence is examined and demonstrated. The proposed scheme has a uniform convergence order of Mx−1+(Δt)2−γ. Two numerical illustrations for various values of ɛ are taken into consideration to validate the theoretical analysis of the scheme.The constructed scheme provides a better order of convergence and more accurate results than methods found in the literature.

Existence of solutions for anisotropic parabolic Ni-Serrin type equations originated from a capillary phenomena with nonstandard growth nonlinearity

We consider an initial boundary value problem for a class of anisotropic parabolic Ni-Serrin type equations with nonstandard nonlinearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. Because the nonlinear perturbation leads to difficulties (it does not have a definite sign) in obtaining a priori estimates in the energy method, we had to modify the Tartar method significantly. Under suitable assumptions, we obtain the global existence, decay and extinction of solutions.

Existence and extinction of solutions for parabolic equations with nonstandard growth nonlinearity

In this paper, we consider an initial boundary value problem for a class of $p(\cdot )$-Laplacian parabolic equation with nonstandard nonlinearity in a bounded domain. By using new approach, we obtain the global and decay of existence of the solutions. Moreover, the precise decay estimates of solutions before the occurrence of the extinction are derived.

Qualitative properties of solutions for dual parabolic equation involving uniformly elliptic nonlocal operator

In this research, we study certain characteristics of the solution for an equation involving uniformly elliptic nonlocal operator by establishing various maximum principles in bounded and unbounded regions. Additionally, we don't need the conditions that the narrow region principle's bound assumption and decay in the unbounded domain. In order to get the monotonicity of the solution, we use several maximum principles and averaging effects to overcome the challenges that generated by the uniformly elliptic nonlocal operator and fractional‐order variable . Of course, while investigating the qualitative characteristics of the equation, we also provide some estimates of auxiliary function. For the study of additional nonlocal operators, we believe the method of uniformly elliptic nonlocal operators will also be helpful.

Boundedness of Solutions of Nonautonomous Degenerate Logistic Equations

AbstractIn this work we analyze the boundedness properties of the solutions of a nonautonomous parabolic degenerate logistic equation in a bounded domain. The equation is degenerate in the sense that the logistic nonlinearity vanishes in a moving region, K(t), inside the domain. The boundedness character of the solutions depends not only on, roughly speaking, the first eigenvalue of the Laplace operator in K(t) but also on the way this moving set evolves inside the domain and in particular on the speed at which it moves.

Monotone Positive Radial Solution of Double Index Logarithm Parabolic Equations

This article mainly studies the double index logarithmic nonlinear fractional g-Laplacian parabolic equations with the Marchaud fractional time derivatives ∂tα. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional g-Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional g-Laplacian parabolic equations is studied.

Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator

Abstract In this article, we consider the parabolic equations with nonlocal Monge-Ampère operators ∂ u ∂ t ( x , t ) − D s θ u ( x , t ) = f ( u ( x , t ) ) , ( x , t ) ∈ R + n × R . \frac{\partial u}{\partial t}\left(x,t)-{D}_{s}^{\theta }u\left(x,t)=f\left(u\left(x,t)),\hspace{1.0em}\left(x,t)\in {{\mathbb{R}}}_{+}^{n}\times {\mathbb{R}}. We first prove the narrow region principle and maximal principle for antisymmetric functions, under the condition that u u is uniformly bounded, which weaken the general decay condition u → 0 u\to 0 at infinity. Then, the monotonicity of positive solutions is established using the method of moving planes.

Large-time behavior of bounded radial solutions of parabolic equations on $${\mathbb R}^{N}$$: Part II—convergence for initial data with a linearly stable limit at infinity

Large-time behavior of bounded radial solutions of parabolic equations on $${\mathbb R}^{N}$$: Part II—convergence for initial data with a linearly stable limit at infinity

Feynman–Kac formula for parabolic Anderson model in Gaussian potential and fractional white noise

In this paper, we establish a Feynman–Kac formula for the stochastic parabolic Anderson model with Gaussian potential in space and fractional white noise in time with Hurst parameter H &amp;gt; 1/2. We obtain the necesscary and suffcient condition for the integrability of the Gaussian potential and the exponential integrability of the solution which is defined by Feynman–Kac formula. By the smoothing of the fractional white noise and techniques from Malliavin calculus, we prove that the Feynman–Kac representation is a mild solution of the stochastic parabolic Anderson equation.

New fractional solutions for the Clannish Random Walker’s Parabolic equation and the Ablowitz-Kaup-Newell-Segur equation

In this article, we find fractional solutions for the Clannish Random Walkers Parabolic (CRWP) equation and Ablowitz-Kaup-Newell-Segeur (AKNS) equation. Different areas of applied sciences, the study of these equations plays a significant role. We select two methods for this purpose, firstly new recent well-established technique is improved Auxiliary equation scheme and second technique is (G′G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\frac{G^{'}}{G^{2}})$$\\end{document}-technique with the help of beta derivative. We used fractional term which involves in beta derivative, in our selected mathematical models. To show how the obtained solutions physically look, they plotted different profiles including both 2D and 3D graphics.

Global existence and blow-up of weak solutions for a fourth-order parabolic equation with gradient nonlinearity

Global existence and blow-up of weak solutions for a fourth-order parabolic equation with gradient nonlinearity

On anisotropic parabolic equation with nonstandard growth order

ABSTRACT In this paper, the existence and the uniqueness of an evolutionary anisotropic p i ( x ) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic p ( x ) − Laplace equation with convection term via L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc.

Analysis of a fourth-order compact $ \theta $-method for delay parabolic equations

&lt;abstract&gt;&lt;p&gt;The upper bounds for the powers of the iteration matrix derived via a numerical method are intimately related to the stability analysis of numerical processes. In this paper, we establish upper bounds for the norm of the &lt;italic&gt;n&lt;/italic&gt;th power of the iteration matrix derived via a fourth-order compact $ \theta $-method to obtain the numerical solutions of delay parabolic equations, and thus present conclusions about the stability properties. We prove that, under certain conditions, the numerical process behaves in a stable manner within its stability region. Finally, we illustrate the theoretical results through the use of several numerical experiments.&lt;/p&gt;&lt;/abstract&gt;

Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

Nonnegative weak solutions of anisotropic parabolic equations

Nonnegative weak solutions of anisotropic parabolic equations

On bounded radial solutions of parabolic equations on $ {\mathbb R}^{N} $: Quasiconvergence for initial data with a stable limit at infinity

On bounded radial solutions of parabolic equations on $ {\mathbb R}^{N} $: Quasiconvergence for initial data with a stable limit at infinity

Quantitative unique continuation for parabolic equations with Neumann boundary conditions

In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator estimates and a global frequency function argument, which is motivated by recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.

Justification of the exact asymptotics of the fundamental solution for a degenerate parabolic equation with a small parameter

AbstractIn this paper, we propose a scheme for constructing the asymptotics of the fundamental solution of a linear degenerate parabolic equation with a small parameter. The asymptotics is constructed using the operator representation of the Dirac delta function and the non‐oscillating Wentzel‐Kramers‐Brillouin (WKB) method. In addition, a method to justify the obtained asymptotics by proving the convergence of the formal series that arises when using the WKB method is shown. Several examples of degenerate parabolic equations are considered, for which a fundamental solution is constructed, and its main properties are indicated.

SIMULATION OF THE PROPAGATION OF A PLANE ELECTROMAGNETIC WAVE IN AN INHOMOGENEOUS NONABSORBING MEDIUM

Analytical solutions of the Ishimaru's parabolic equation for the coherence function of the electromagnetic field, which describe the temporal properties of the pulse at the output of a heterogeneous non-dissipative medium, are considered. A generalization of the approach used in the Ishimaru model to describe the time evolution of a monochromatic electromagnetic pulse enveloping in homogeneous non-dissipative media is obtained for the case of non-homogeneous non-dissipative media. Thus, an attempt was made to take into account the influence of the heterogeneity of the medium on the shape of the resulting pulse. When solving the given problem, the difficulties associated with the calculation of the continuous integral arising in the space of diffusion trajectories were overcome. This made it possible to obtain an explicit expression for the Green's function of the task and to build a computational algorithm based on which a number of numerous experiments were conducted. The analysis of the work was carried out based on the apparatus of quadratic integral functionals based on the solutions of differential stochastic equations. In the paper, the invariant temporal properties of the envelope of monochromatic electromagnetic pulses recorded after passing through a flat layer of a scattering heterogeneous medium, i.e., properties that remain unchanged when the parameters of the medium vary, in particular, the distribution of the concentration of scattering centers. The dynamics of the formation of time plumes of the scattered wave, in which the tail part is located in the peripheral time area, were analyzed. When propagating at the speed of transformation of the wave front, it reflects the appearance of the layers of the scattering region and its longitudinal shape. It is noted that the development of the proposed approximation approach to the processes affecting the time delay of electromagnetic pulses can be the accounting for the attenuation of radiation during its propagation in an inhomogeneous absorbing medium.

On the existence of time-periodic solutions of nonlinear parabolic differential equations with nonlocal boundary conditions of the Bitsadze-Samarskii type

We study a nonlinear parabolic differential equation in a bounded multidimensional domain with nonlocal boundary conditions of the Bitsadze-Samarskii type. We prove existence theorems for a periodic in time generalized solution. Su cient conditions for the existence of generalized solutions contain either an algebraic ellipticity condition or an algebraic strong ellipticity condition for the auxiliary differential-difference operator.