Nonlinear equations with degenerate operator at fractional Caputo derivative

2016 ◽  
Vol 40 (17) ◽  
pp. 6138-6146 ◽  
Author(s):  
Marina V. Plekhanova
Keyword(s):  
Nonlinear Equations ◽  
Caputo Derivative ◽  
Fractional Caputo Derivative ◽  
Degenerate Operator

scholarly journals The Third Boundary Value Problem for a Loaded Thermal Conductivity Equation with a Fractional Caputo Derivative

2020 ◽  
pp. 52-64
Author(s):  
M. Kh. Beshtokov ◽  
M. Z. Khudalov
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Caputo Derivative ◽  
Boundary Value ◽  
Conductivity Equation ◽  
Partial Differential ◽  
The Third ◽  
Fractional Caputo Derivative

Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional (porous media) orders are widely used, since analytical solving methods for solving are impossible.In the paper, we study the initial-boundary value problem for the loaded differential heat equation with a fractional Caputo derivative and conditions of the third kind. To solve the problem on the assumption that there is an exact solution in the class of sufficiently smooth functions by the method of energy inequalities, a priori estimates are obtained both in the differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. Due to the linearity of the problem under consideration, these inequalities allow us to state the convergence of the approximate solution to the exact solution at a rate equal to the approximation order of the difference scheme. An algorithm for the numerical solution of the problem is constructed.

scholarly journals Approximate solutions of one-dimensional systems with fractional derivative

2020 ◽  
Vol 31 (07) ◽  
pp. 2050092
Author(s):  
A. Ferrari ◽  
M. Gadella ◽  
L. P. Lara ◽  
E. Santillan Marcus
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Derivative ◽  
Approximate Solutions ◽  
One Dimensional ◽  
Dissipative Term ◽  
Numerical Resolution ◽  
Fractional Caputo Derivative ◽  
Fractional Differential

The fractional calculus is useful to model nonlocal phenomena. We construct a method to evaluate the fractional Caputo derivative by means of a simple explicit quadratic segmentary interpolation. This method yields to numerical resolution of ordinary fractional differential equations. Due to the nonlocality of the fractional derivative, we may establish an equivalence between fractional oscillators and ordinary oscillators with a dissipative term.

A fractional model of a dynamical Brusselator reaction-diffusion system arising in triple collision and enzymatic reactions

2016 ◽  
Vol 5 (4) ◽  
Author(s):  
Jagdev Singh ◽  
M.M. Rashidi ◽  
Devendra Kumar ◽  
Ram Swroop
Keyword(s):  
Present Article ◽  
Caputo Derivative ◽  
Reaction Diffusion ◽  
Triple Collision ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Enzymatic Reactions ◽  
Fractional Model ◽  
Fractional Caputo Derivative

AbstractIn this paper, we study a dynamical Brusselator reaction-diffusion system arising in triple collision and enzymatic reactions with time fractional Caputo derivative. The present article involves a more generalized effective approach, proposed for the Brusselator system say

scholarly journals A PRIORI ESTIMATION OF A GENERALIZED NONLOCAL BOUNDARY VALUE PROBLEM FOR A THRID ORDER EQUATION WITH A FRACTIONAL TIME CAPUTO DERIVATIVE

2019 ◽  
pp. 35-40
Author(s):  
А.М. Шхагапсоев
Keyword(s):  
Boundary Value Problem ◽  
Caputo Derivative ◽  
Boundary Value ◽  
A Priori ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
Fractional Caputo Derivative ◽  
Multiple Characteristics ◽  
A Priori Estimation

Рассматривается краевая задача для уравнения третьего порядка параболического типа с дробной производной Капуто. Методом энергетических неравенств получена априорная оценка решения обобщенной нелокальной краевой задачи для уравнения с кратными характеристиками с дробной производной Капуто по времени. A boundary value problem for a third-order parabolic equation with a fractional Caputo derivative is considered. A priori estimation of the solution of a generalized nonlocal boundary value problem for an equation with multiple characteristics with a fractional Caputo derivative in time is obtained by the method of energy inequalities.

scholarly journals Fractional advection differential equation within Caputo and Caputo–Fabrizio derivatives

2016 ◽  
Vol 8 (12) ◽  
pp. 168781401668330 ◽  
Author(s):  
Dumitru Baleanu ◽  
Bahram Agheli ◽  
Maysaa Mohamed Al Qurashi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Perturbation Method ◽  
Caputo Derivative ◽  
Detailed Comparison ◽  
Analysis Method ◽  
Partial Differential ◽  
Fractional Caputo Derivative

In this research, we applied the variational homotopic perturbation method and q-homotopic analysis method to find a solution of the advection partial differential equation featuring time-fractional Caputo derivative and time-fractional Caputo–Fabrizio derivative. A detailed comparison of the obtained results was reported. All computations were done using Mathematica.

Cauchy formula for the time-varying linear systems with caputo derivative

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Tadeusz Kaczorek ◽  
Dariusz Idczak
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Caputo Derivative ◽  
Cauchy Formula ◽  
Time Varying ◽  
Fractional Caputo Derivative ◽  
Standard Relation

AbstractIn the paper, existence, uniqueness and a Cauchy formula for the solution to a time-varying linear system containing fractional Caputo derivative is obtained. This formula shows that nonnegativity of the data of the system implies nonnegativity of the solution. In the context of a strenghthening of this result, an example illustrating the absence (in the case of Caputo derivative) of the standard relation “monotonicity of function - sign of derivative”.

Fractional investigation of bank data with fractal-fractional Caputo derivative

2020 ◽  
Vol 131 ◽  
pp. 109528 ◽  
Author(s):  
Zhongfei Li ◽  
Zhuang Liu ◽  
Muhammad Altaf Khan
Keyword(s):  
Caputo Derivative ◽  
Fractional Caputo Derivative

scholarly journals Existence and regularity results for stochastic fractional pseudo-parabolic equations driven by white noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tran Ngoc Thach ◽  
Devendra Kumar ◽  
Nguyen Hoang Luc ◽  
Nguyen Huy Tuan
Keyword(s):  
Parabolic Equation ◽  
White Noise ◽  
Linear Space ◽  
Parabolic Equations ◽  
Caputo Derivative ◽  
Direct Problem ◽  
Mild Solutions ◽  
Fractional Caputo Derivative ◽  
Existence And Regularity Results ◽  
Regularity Results

<p style='text-indent:20px;'>Solutions of a direct problem for a stochastic pseudo-parabolic equation with fractional Caputo derivative are investigated, in which the non-linear space-time-noise is assumed to satisfy distinct Lipshitz conditions including globally and locally assumptions. The main aim of this work is to establish some existence, uniqueness, regularity, and continuity results for mild solutions.</p>

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