FEATURES OF SEEPAGE OF A LIQUID TO A CHINK IN THE CRACKED DEFORMABLE LAYER

2010 ◽  
Vol 01 (03) ◽  
pp. 333-347 ◽  
Author(s):  
T. S. ALEROEV ◽  
H. T. ALEROEVA ◽  
JIANFEI HUANG ◽  
NINGMING NIE ◽  
YIFA TANG ◽  
...  
Keyword(s):  
Differential Equation ◽  
Initial Value Problem ◽  
Numerical Scheme ◽  
Variable Coefficients ◽  
Initial Value ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Model ◽  
New Model ◽  
Whole Process ◽  
Fractional Differential

We establish a new model for seepage of a liquid to a chink in the cracked deformable layer, an initial value problem of nonlinear fractional differential equation with variable coefficients, then design a numerical scheme of order 2 to solve this initial value problem. This new model theoretically explains the operating thickness H of a layer depending on the values of pressure gradient on the whole chink rather than on one point, which is practiced by a large amount of data. Compared with the Dontsov equation, our fractional model considers more aspects of the whole process. The earlier rejected results can also be considered in the display lines of the fractional model.

scholarly journals Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Qiuping Li ◽  
Shurong Sun ◽  
Ping Zhao ◽  
Zhenlai Han
Keyword(s):  
Differential Equation ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Upper And Lower Solutions ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Nonlinear Fractional Differential Equation ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential

We discuss the initial value problem for the nonlinear fractional differential equationL(D)u=f(t,u),  t∈(0,1],  u(0)=0, whereL(D)=Dsn-an-1Dsn-1-⋯-a1Ds1,0<s1<s2<⋯<sn<1, andaj<0,j=1,2,…,n-1,Dsjis the standard Riemann-Liouville fractional derivative andf:[0,1]×ℝ→ℝis a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.

scholarly journals Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
D. Baleanu ◽  
H. Mohammadi ◽  
Sh. Rezapour
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Initial Value Problem ◽  
Initial Value ◽  
Nonlinear Fractional Differential Equation ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

We investigate the existence and multiplicity of positive solutions for the nonlinear fractional differential equation initial value problemD0+αu(t)+D0+βu(t)=f(t,u(t)), u(0)=0, 0<t<1, where0<β<α<1, D0+αis the standard Riemann-Liouville differentiation andf:[0,1]×[0,∞)→[0,∞)is continuous. By using some fixed-point results on cones, some existence and multiplicity results of positive solutions are obtained.

scholarly journals Positive Solution of a Nonlinear Fractional Differential Equation Involving Caputo Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Changyou Wang ◽  
Haiqiang Zhang ◽  
Shu Wang
Keyword(s):  
Differential Equation ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Caputo Derivative ◽  
Nonlinear Term ◽  
Upper And Lower Solutions ◽  
Initial Value ◽  
Nonlinear Fractional Differential Equation ◽  
Control Functions ◽  
Fractional Differential

This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained.

Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Myong-Ha Kim ◽  
Guk-Chol Ri ◽  
Hyong-Chol O
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
Operational Method ◽  
Initial Value ◽  
Constant Coefficients ◽  
Generalized Fractional Derivatives ◽  
Fractional Differential

AbstractThis paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski’s type. We prove that the initial value problem has the solution if and only if some initial values are zero.

scholarly journals Initial Value Problem for Stochastic Hyprid Hadamard Fractional Differential Equation

2019 ◽  
Vol 16 ◽  
pp. 8288-8296
Author(s):  
Mahmoud Mohammed Mostafa El-Borai ◽  
Wagdy G. El-sayed ◽  
A. A. Badr ◽  
Ahmed Tarek Sayed
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Initial Value Problem ◽  
Stochastic Analysis ◽  
Fractional Integral ◽  
Existence Of Solutions ◽  
Initial Value ◽  
Analysis Techniques ◽  
Fractional Differential ◽  
Hadamard Fractional Integral

In this paper, we discuss the existence of solutions for a stochastic initial value problem of Hyprid fractional dierential equations of Hadamard type given by                            where HD is the Hadamard fractional derivative, and is the Hadamard fractional integral and be such that are investigated. The fractional calculus and stochastic analysis techniques are used to obtain the required results. 

scholarly journals Solution of a Class of Differential Equation with Variable Coefficients

2016 ◽  
Vol 8 (4) ◽  
pp. 140
Author(s):  
Huanhuan Xiong ◽  
Yuedan Jin ◽  
Xiangqing Zhao
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Value Problem ◽  
Variable Coefficient ◽  
Variable Coefficients ◽  
Hyperbolic Partial Differential Equation ◽  
Initial Value ◽  
Partial Differential

<p>In this paper, we obtain the formula of solution to the initial value problem for a hyperbolic partial differential equation with variable coefficient which is the modification of the famous D’ Alembert formula.</p>

scholarly journals On a Hilfer Type Fractional Differential Equation with Nonlinear Right-Hand Side

2021 ◽  
Vol 103 (3) ◽  
pp. 140-155
Author(s):  
T. K. Yuldashev ◽  
◽  
B. J. Kadirkulov ◽  
A. R. Marakhimov ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Initial Value Problem ◽  
Integral Transformation ◽  
Numerical Realization ◽  
Initial Value ◽  
Volterra Type ◽  
Fractional Integral Equation ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

In this article we consider the questions of one-valued solvability and numerical realization of initial value problem for a nonlinear Hilfer type fractional differential equation with maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation. The theorem of existence and uniqueness of the solution of given initial value problem in the segment under consideration is proved. For numerical realization of solution the generalized Jacobi–Galerkin method is applied. Illustrative examples are provided.

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