scholarly journals Diffusion-Wave Type Solutions to the Second-Order Evolutionary Equation with Power Nonlinearities

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 232
Author(s):  
Alexander Kazakov ◽  
Anna Lempert
Keyword(s):  
Porous Medium Equation ◽  
Second Order ◽  
Phase Portraits ◽  
Evolutionary Equation ◽  
Finite Difference Schemes ◽  
Cauchy Problems ◽  
Wave Type ◽  
Diffusion Wave ◽  
Medium Equation ◽  
Special Cases

The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous medium equation and its generalizations. We construct solutions that describe perturbations propagating over a zero background with a finite velocity. Such effects are known to be atypical for parabolic equations and appear as a consequence of the degeneration of the equation at the points where the desired function vanishes. Previously, we have constructed it, but here the case of power nonlinearity is considered. It allows for conducting a more detailed analysis. We prove a new theorem for the existence of solutions of this type in the class of piecewise analytical functions, which generalizes and specifies the earlier statements. We find and study exact solutions having the diffusion wave type, the construction of which is reduced to the second-order Cauchy problem for an ordinary differential equation (ODE) that inherits singularities from the original formulation. Statements that ensure the existence of global continuously differentiable solutions are proved for the Cauchy problems. The properties of the constructed solutions are studied by the methods of the qualitative theory of differential equations. Phase portraits are obtained, and quantitative estimates are determined by constructing and analyzing finite difference schemes. The most significant result is that we have shown that all the special cases for incomplete equations take place for the complete equation, and other configurations of diffusion waves do not arise.

scholarly journals Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 871
Author(s):  
Alexander Kazakov
Keyword(s):  
Parabolic Equations ◽  
Unknown Function ◽  
Heat Waves ◽  
Parabolic Type ◽  
Porous Medium Equation ◽  
Heat Propagation ◽  
Second Order ◽  
Cauchy Problems ◽  
Medium Equation ◽  
Special Cases

The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity.

scholarly journals On the Solutions of a Porous Medium Equation with Exponent Variable

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

The paper studies the initial-boundary value problem of a porous medium equation with exponent variable. How to deal with nonlinear term with the exponent variable is the main dedication of this paper. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of weak solutions is studied. In some special cases, the stability of weak solutions can be proved without any boundary value condition.

scholarly journals Rate of convergence and error estimates for finite-difference schemes of solving linear ill-posed Cauchy problems of the second order

2017 ◽  
pp. 322-347
Author(s):  
М.М. Кокурин
Keyword(s):  
Finite Difference ◽  
Rate Of Convergence ◽  
Error Estimates ◽  
Sufficient Conditions ◽  
Second Order ◽  
Difference Schemes ◽  
Order Differential Operator ◽  
Finite Difference Schemes ◽  
Cauchy Problems ◽  
Ill Posed

Изучаются конечно-разностные схемы решения некорректных задач Коши для линейного дифференциально-операторного уравнения второго порядка в банаховом пространстве. Получены равномерные по времени оценки скорости сходимости и погрешности этих схем при наложении на искомое решение условия истокопредставимости. Найдены близкие друг к другу необходимые и достаточные условия в терминах показателя истокопредставимости для сходимости класса схем со степенной скоростью относительно шага дискретизации. Построены и изучены схемы полной дискретизации некорректных задач Коши второго порядка, сочетающие полудискретизацию по времени с дискретной аппроксимацией пространства и оператора. Finite-difference schemes of solving ill-posed Cauchy problems for linear second-order differential operator equations in Banach spaces are considered. Several time-uniform rate of convergence and error estimates are obtained for the considered schemes under the assumption that the sought solution satisfies the sourcewise condition. Necessary and sufficient conditions are found in terms of sourcewise index for a class of schemes with the power convergence rate with respect to the discretization step. A number of full discretization schemes for second-order ill-posed Cauchy problems are proposed on the basis of combining the half-discretization in time with the discrete approximation of the spaces and the operators.

scholarly journals The Stability of the Solutions for a Porous Medium Equation with a Convection Term

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Huashui Zhan ◽  
Miao Ouyang
Keyword(s):  
Porous Medium ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convection Term ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

This paper studies the initial-boundary value problem of a porous medium equation with a convection term. If the equation is degenerate on the boundary, then only a partial boundary condition is needed generally. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of the solutions is studied. In some special cases, the stability can be proved without any boundary value condition.

Harnack estimates for a kind of porous medium equation

2021 ◽  
Vol 115 ◽  
pp. 106978
Author(s):  
Feida Jiang ◽  
Xinyi Shen ◽  
Hui Wu
Keyword(s):  
Porous Medium ◽  
Porous Medium Equation ◽  
Medium Equation ◽  
Harnack Estimates

scholarly journals Global existence of solutions and smoothing effects for classes of reaction–diffusion equations on manifolds

2021 ◽  
Author(s):  
Gabriele Grillo ◽  
Giulia Meglioli ◽  
Fabio Punzo
Keyword(s):  
Porous Medium ◽  
Global Existence ◽  
Source Term ◽  
Reaction Diffusion ◽  
Porous Medium Equation ◽  
Variable Density ◽  
Reaction Diffusion Equations ◽  
Medium Equation ◽  
Global Existence Of Solutions ◽  
Smoothing Effects

AbstractWe consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on p and m in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequalities hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in $${{\mathbb {R}}}^n$$ R n .

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