Three solutions for a class of fractional impulsive advection‐dispersion equations with Sturm‐Liouville boundary conditions via variational approach

2020 ◽  
Vol 43 (15) ◽  
pp. 9151-9168
Author(s):  
Dandan Min ◽  
Fangqi Chen
Keyword(s):  
Boundary Conditions ◽  
Variational Approach ◽  
Three Solutions ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Sturm Liouville

Variational approaches to systems of Sturm–Liouville boundary value problems

2020 ◽  
pp. 2150032
Author(s):  
Shapour Heidarkhani ◽  
Ghasem A. Afrouzi ◽  
Shahin Moradi
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Technical Approach ◽  
Three Solutions ◽  
Value System ◽  
Sturm Liouville ◽  
Variational Approaches

In this paper, we consider the existence of one solution and three solutions for the boundary value system with Sturm–Liouville boundary conditions [Formula: see text] for [Formula: see text]. Our technical approach is based on variational methods. In addition, examples are provided to illustrate our results.

scholarly journals A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Taohua Liu ◽  
Muzhou Hou
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Difference Method ◽  
Computational Cost ◽  
First Order ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Implicit Finite Difference

Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K) and computational cost of O(Klog⁡K). Traditionally, the Gaussian elimination method requires storage of O(K2) and computational cost of O(K3). Finally, the accuracy and efficiency of the method are checked with a numerical example.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

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