scholarly journals On a discrete scheme for time fractional fully nonlinear evolution equations

2020 ◽  
Vol 120 (1-2) ◽  
pp. 151-162 ◽  
Author(s):  
Yoshikazu Giga ◽  
Qing Liu ◽  
Hiroyoshi Mitake
Keyword(s):  
Fractional Derivatives ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Parabolic Pdes ◽  
Fully Nonlinear ◽  
Discrete Scheme ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Convergence Of The Scheme

We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo’s time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can be viewed as a resolvent-type approximation.

scholarly journals A Novel Analytical Technique to Obtain Kink Solutions for Higher Order Nonlinear Fractional Evolution Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Qazi Mahmood Ul Hassan ◽  
Jamshad Ahmad ◽  
Muhammad Shakeel
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Fractional Evolution Equations ◽  
Analytical Technique ◽  
Fractional Complex Transform ◽  
Fifth Order

We use the fractional derivatives in Caputo’s sense to construct exact solutions to fractional fifth order nonlinear evolution equations. A generalized fractional complex transform is appropriately used to convert this equation to ordinary differential equation which subsequently resulted in a number of exact solutions.

scholarly journals A probabilistic numerical method for fully nonlinear parabolic PDEs

2011 ◽  
Vol 21 (4) ◽  
pp. 1322-1364 ◽  
Author(s):  
Arash Fahim ◽  
Nizar Touzi ◽  
Xavier Warin
Keyword(s):  
Numerical Method ◽  
Parabolic Pdes ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic

Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs

2015 ◽  
Vol 18 (5) ◽  
pp. 1482-1503 ◽  
Author(s):  
Tao Kong ◽  
Weidong Zhao ◽  
Tao Zhou
Keyword(s):  
Backward Stochastic Differential Equation ◽  
High Order ◽  
Cauchy Problems ◽  
Numerical Schemes ◽  
Parabolic Pdes ◽  
Hjb Equations ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Nonlinear Parabolic Pde

AbstractIn this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.

scholarly journals On some nonlinear parabolic equations with variable exponents and measure data

2020 ◽  
Vol 6 (1) ◽  
pp. 93-117
Author(s):  
Bouchra El Hamdaoui ◽  
Jaouad Bennouna
Keyword(s):  
Parabolic Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Parabolic Equations ◽  
Dirichlet Condition ◽  
Nonlinear Evolution Equations ◽  
Measure Data ◽  
Renormalized Solutions ◽  
Generalized Sobolev Spaces ◽  
Nonlinear Parabolic

AbstractWe prove the existence of renormalized solutions to a class of nonlinear evolution equations, supplemented with initial and Dirichlet condition in the framework of generalized Sobolev spaces. The data are assumed merely integrable.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

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