scholarly journals A single quenching point for a fractional heat equation based on the Riemann-Liouville fractional derivative with a nonlinear concentrate source

2017 ◽  
Vol 2017 (1) ◽  
Author(s):  
Wannika Sawangtong ◽  
Panumart Sawangtong
Keyword(s):  
Heat Equation ◽  
Fractional Derivative ◽  
Fractional Heat Equation ◽  
Liouville Fractional Derivative ◽  
Concentrate Source

Similarity solutions and the computation formulas of a nonlinear fractional-order generalized heat equation

2019 ◽  
Vol 33 (10) ◽  
pp. 1950122 ◽  
Author(s):  
Yufeng Zhang
Keyword(s):  
Heat Equation ◽  
Fractional Derivative ◽  
Numerical Solutions ◽  
Similarity Solutions ◽  
Nonlinear Heat Equation ◽  
Order Ordinary Differential Equation ◽  
Nonlinear Heat Conduction Equation ◽  
Liouville Fractional Derivative ◽  
Generalized Heat Equation ◽  
Two Parameters

A generalized nonlinear heat equation with the fractional derivative is proposed, whose similarity solutions are derived from a type of special scalar transformation with two parameters. With the help of separated variable method, two special series solutions of the standard heat equation are obtained. Finally, through computation of the left Riemann–Liouville fractional derivative, we obtain two approximated computation formulas of the factional-order ordinary differential equation which could be used to calculate the numerical solutions of the generalized nonlinear heat conduction equation.

scholarly journals The solutions of time and space conformable fractional heat equations with conformable Fourier transform

2015 ◽  
Vol 7 (2) ◽  
pp. 130-140 ◽  
Author(s):  
Yücel Çenesiz ◽  
Ali Kurt
Keyword(s):  
Fourier Transform ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Heat Equations ◽  
Conformable Fractional Derivative ◽  
Time And Space ◽  
Fourier Cosine ◽  
Fractional Heat Equation ◽  
Fourier Cosine Transform ◽  
Definition Of

Abstract In this paper our aim is to find the solutions of time and space fractional heat differential equations by using new definition of fractional derivative called conformable fractional derivative. Also based on conformable fractional derivative definition conformable Fourier Transform is defined. Fourier sine and Fourier cosine transform definitions are given and space fractional heat equation is solved by conformable Fourier transform.

scholarly journals Solution of One Dimensional Conformable fractional Heat Equation

2020 ◽  
Vol 9 (5) ◽  
pp. 693-695
Keyword(s):  
Heat Equation ◽  
Fractional Derivative ◽  
Conformable Fractional Derivative ◽  
One Dimensional ◽  
Fractional Heat Equation ◽  
Definition Of

The aim of this paper is to present the solution of one dimensional conformable fractional heat equation by applying conformable fractional derivative which is considered as more convenient definition of fractional derivative.

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

Optical soliton solutions of the time-fractional perturbed Fokas-Lenells equation: Riemann-Liouville fractional derivative

Optik ◽  
2019 ◽  
Vol 183 ◽  
pp. 1114-1119 ◽  
Author(s):  
Bashir Salah ◽  
Essam R. El-Zahar ◽  
A.F. Aljohani ◽  
A. Ebaid ◽  
Mohammed Krid
Keyword(s):  
Fractional Derivative ◽  
Soliton Solutions ◽  
Optical Soliton ◽  
Liouville Fractional Derivative

NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341001 ◽  
Author(s):  
LEEVAN LING ◽  
MASAHIRO YAMAMOTO
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Long Time ◽  
Liouville Fractional Derivative ◽  
Time Fractional Diffusion Equation

We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.

scholarly journals Hardy-Type Inequalities for an Extension of the Riemann- Liouville Fractional Derivative Operators

2021 ◽  
Vol 45 (5) ◽  
pp. 797-813
Author(s):  
SAJID IQBAL ◽  
◽  
GHULAM FARID ◽  
JOSIP PEČARIĆ ◽  
ARTION KASHURI
Keyword(s):  
Fractional Derivative ◽  
Convex Functions ◽  
Mean Value ◽  
Mean Value Theorems ◽  
Hardy Type Inequalities ◽  
Exponential Convexity ◽  
Hardy Type ◽  
Liouville Fractional Derivative ◽  
Fractional Derivative Operators

In this paper we present variety of Hardy-type inequalities and their refinements for an extension of Riemann-Liouville fractional derivative operators. Moreover, we use an extension of extended Riemann-Liouville fractional derivative and modified extension of Riemann-Liouville fractional derivative using convex and monotone convex functions. Furthermore, mean value theorems and n-exponential convexity of the related functionals is discussed.

scholarly journals Determination of source term for fractional heat equation on the sphere

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 361-370
Author(s):  
Nguyen Phuong ◽  
Tran Binh ◽  
Nguyen Luc ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Fractional Heat Equation ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

In this work, we study a truncation method to solve a time fractional diffusion equation on the sphere of an inverse source problem which is ill-posed in the sense of Hadamard. Through some priori assumption, we present the error estimates between the regularized and exact solutions.

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