Fractional Sturm-Liouville Problem and 1D Space-Time Fractional Diffusion With Mixed Boundary Conditions

2015 ◽  
Author(s):  
Malgorzata Klimek
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Liouville Problem ◽  
Mixed Boundary Conditions ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Space Time ◽  
Sturm Liouville ◽  
Derivatives Of ◽  
Space Interval

In the paper, we show a connection between a regular fractional Sturm-Liouville problem with left and right Caputo derivatives of order in the range (1/2, 1) and a 1D space-time fractional diffusion problem in a bounded domain. Both problems include mixed boundary conditions in a finite space interval. We prove that in the case of vanishing mixed boundary conditions, the Sturm-Liouville problem can be rewritten in terms of Riesz derivatives. Then, we apply earlier results on its eigenvalues and eigenfunctions to construct a weak solution of the 1D fractional diffusion equation with variable diffusivity. Adding an assumption on the summability of the eigenvalues’ inverses series, we formulate a theorem on a strong solution of the 1D fractional diffusion problem.

scholarly journals Complete Solution For The Time Fractional Diffusion Problem With Mixed Boundary Conditions by Operational Method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Aghili
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Heat Conduction Problem ◽  
Complete Solution ◽  
Mixed Boundary Conditions ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Two Dimensions ◽  
Operational Method

AbstractIn this study, we present some new results for the time fractional mixed boundary value problems. We consider a generalization of the Heat - conduction problem in two dimensions that arises during the manufacturing of p - n junctions. Constructive examples are also provided throughout the paper. The main purpose of this article is to present mathematical results that are useful to researchers in a variety of fields.

scholarly journals Exact and numerical solutions of the fractional Sturm–Liouville problem

2018 ◽  
Vol 21 (1) ◽  
pp. 45-71 ◽  
Author(s):  
Malgorzata Klimek ◽  
Mariusz Ciesielski ◽  
Tomasz Blaszczyk
Keyword(s):  
Boundary Conditions ◽  
Numerical Method ◽  
Rate Of Convergence ◽  
Numerical Solutions ◽  
Mixed Boundary ◽  
Liouville Problem ◽  
Mixed Boundary Conditions ◽  
Experimental Rate ◽  
New Variant ◽  
Sturm Liouville

AbstractIn the paper, we discuss the regular fractional Sturm-Liouville problem in a bounded domain, subjected to the homogeneous mixed boundary conditions. The results on exact and numerical solutions are based on transformation of the differential fractional Sturm-Liouville problem into the integral one. First, we prove the existence of a purely discrete, countable spectrum and the orthogonal system of eigenfunctions by using the tools of Hilbert-Schmidt operators theory. Then, we construct a new variant of the numerical method which produces eigenvalues and approximate eigenfunctions. The convergence of the procedure is controlled by using the experimental rate of convergence approach and the orthogonality of eigenfunctions is preserved at each step of approximation. In the final part, the illustrative examples of calculations and estimation of the experimental rate of convergence are presented.

A VISCOUS-INVISCID COUPLING UNDER MIXED BOUNDARY CONDITIONS

1992 ◽  
Vol 02 (04) ◽  
pp. 461-482 ◽  
Author(s):  
C. CANUTO ◽  
A. RUSSO
Keyword(s):  
Boundary Conditions ◽  
Original Problem ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Diffusion Problem ◽  
Convection Diffusion ◽  
Convection Equation ◽  
Convection Dominated

In this paper we consider a nonlinear modification of a linear convection-diffusion problem in order to get a pure convection equation where the original problem is convection dominated. We extend the results of previous papers by considering mixed Dirichlet/Oblique derivative boundary conditions.

Solution to a diffusion problem with mixed boundary conditions

1974 ◽  
Vol 17 (3) ◽  
pp. 315-318 ◽  
Author(s):  
D.I. Cherednichenko ◽  
Harry Gruenberg ◽  
Tapan K. Sarkar
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Diffusion Problem

Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
Małgorzata Klimek ◽  
Agnieszka B. Malinowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Waiting Times ◽  
Liouville Problem ◽  
Fractional Diffusion ◽  
Space Time ◽  
Domain Model ◽  
Diffusion Equations ◽  
Liouville Theory ◽  
Finite Domain ◽  
Fractional Diffusion Equations ◽  
Sturm Liouville

AbstractThe space–time fractional diffusion equations on finite domain model anomalous diffusion behavior with large particle jumps combined with long waiting times. In this work we prove existence of strong solutions for such equations. Our proofs strongly depend on the fractional Sturm–Liouville theory, precisely on the problem of finding eigenvalues and corresponding eigenfunctions to the certain fractional differential equation. Using the method of separating variables and applying theorem ensuring existence of solutions to the fractional Sturm–Liouville problem we solve several types of fractional diffusion equations.

scholarly journals Iterative positive solutions to a coupled fractional differential system with the multistrip and multipoint mixed boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiaodi Zhao ◽  
Yuehan Liu ◽  
Huihui Pang
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Mixed Boundary ◽  
Coupled System ◽  
Mixed Boundary Conditions ◽  
Monotone Iterative Technique ◽  
Monotone Iterative ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
Derivatives Of

Abstract Using the monotone iterative technique, we investigate the existence of iterative positive solutions to a coupled system of fractional differential equations supplemented with multistrip and multipoint mixed boundary conditions. It is worth mentioning that the nonlinear terms of the system depend on the lower fractional-order derivatives of the unknown functions and the boundary conditions involve the combination of the multistrip fractional integral and the multipoint value of the unknown functions in $[0,1]$ [ 0 , 1 ] .

REMARK ON ESTIMATE OF A POTENTIAL IN TERMS OF EIGENVALUES OF THE STURM–LIOUVILLE OPERATOR

2008 ◽  
Vol 22 (23) ◽  
pp. 2177-2180
Author(s):  
EVGENY KOROTYAEV
Keyword(s):  
Boundary Conditions ◽  
Liouville Operator ◽  
A Priori Estimates ◽  
Mixed Boundary ◽  
A Priori ◽  
Mixed Boundary Conditions ◽  
Unit Interval ◽  
Priori Estimates ◽  
Sturm Liouville ◽  
Neumann Eigenvalues

We consider the Sturm–Liouville operator on the unit interval. We obtain two-sided a priori estimates of potential in terms of Dirichlet and Neumann eigenvalues and eigenvalues for 2 types of mixed boundary conditions.

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