Non-existence of Global Solutions to a System of Fractional Diffusion Equations

Kirane, Mokhtar; Ahmad, Bashir; Alsaedi, Ahmed; Al-Yami, Maryem

doi:10.1007/s10440-014-9865-4

Cited by 28 publications

(10 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Consequently, we show that there exists a mild solution u ∈ C((0, T]; L 2 (Ω)) which can be expressed as in (8). Furthermore, together (11), (13), and 17, the estimate (10) is obtained.…”

Section: Definition 31mentioning

confidence: 82%

“…We refer the books 4,6 and the papers in other works. [7][8][9][10][11][12][13] The direct problem of fractional diffusion equations (1) is the typical heat equation for = 1 equipped with A being the Laplacian operator. Such equations were introduced by Schneider and Wyss, 14 and they pointed out that such equations can be described of diffusion in special types of porous media and it can represent the "gray" noise in the fractional diffusion for 0 < < 1 while the white noise for = 1.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Existence and regularity results of a backward problem for fractional diffusion equations

Zhou

Ahmad

et al. 2019

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we study a backward problem for an inhomogeneous fractional diffusion equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence, uniqueness, and regularity of the mild solutions as well as the classical solutions of the proposed problem in a weighted Hölder continuous function space. KEYWORDS backward problem, existence, fractional diffusion equation, regularity MSC CLASSIFICATION 26A33; 35R11 d dt ∫ t 0 (t − s) −1 u(s, x)ds, t > 0, Math Meth Appl Sci. 2019;42:6775-6790.wileyonlinelibrary.com/journal/mma

show abstract

Section: Definition 31mentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Existence and regularity results of a backward problem for fractional diffusion equations

Zhou

Ahmad

et al. 2019

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…It is very diffi cult if not impossible to obtain analytical solutions for most fractional diff erential equations. However, some analytical (Podlubny, 1999;Diethelm, 2010;Gafi ychuk & Datsko, 2010;Kirane et al, 2014) and numerical (Yuste & Acedo, 2005;Zhuang & Liu, 2006;Yu et al, 2008;Rida et al, 2010;Scherer et al, 2011;Xu & He, 2011;Deng & Li, 2012;Deng et al, 2016) techniques play an important role in identifying the solution behaviour and obtaining approximate solutions of such fractional diff erential equations.…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical method for fractional ordinary differential equations based on exponentially decreasing random memory on uniform meshes

Somathilake

2020

J. Natn. Sci. Foundation Sri Lanka

View full text Add to dashboard Cite

This paper proposes a new numerical method to solve non-linear fractional ordinary diff erential equations (FODEs) of the form , with initial conditions. Here, is a continuous function, is an arbitrary positive real number and the fractional diff erential operator, , is in the sense of Caputo derivative. Fixed (short) memory method (SMM) and full memory method (FMM) are two established numerical methods for fractional diff erential equations. In fi xed memory method, tail of the memory at each time step is cut off and hence an uncontrollable error occurs. Further, full memory method is not suitable for long time integration of fractional diff erential equations because of high computational cost. In the proposed method [hereinafter referred to as decreasing random memory method (DRMM)], number of memory points in the past are chosen randomly and they are decreasing along the tail of the memory. Numerical experiments showed that the error occurs in the proposed DRMM is less than that of SMM. The solutions obtained by FMM and DRMM were also very close to the actual solutions of the considered fractional diff erential equations. The proposed method-DRMM is more accurate than the SMM and the estimated order of convergence (EOC) of DRMM is almost same as that of FMM. The proposed method DRMM is more effi cient than the established methods SMM and FMM.

show abstract

“…Some of the latest studies on integral and nonlocal boundary value problems for coupled fractional differential equations are presented in [17][18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

On a Coupled System of Fractional Differential Equations with Four Point Integral Boundary Conditions

2019

View full text Add to dashboard Cite

The study is on the existence of the solution for a coupled system of fractional differential equations with integral boundary conditions. The first result will address the existence and uniqueness of solutions for the proposed problem and it is based on the contraction mapping principle. Secondly, by using Leray–Schauder’s alternative we manage to prove the existence of solutions. Finally, the conclusion is confirmed and supported by examples.

show abstract

Non-existence of Global Solutions to a System of Fractional Diffusion Equations

Cited by 28 publications

References 14 publications

Existence and regularity results of a backward problem for fractional diffusion equations

Existence and regularity results of a backward problem for fractional diffusion equations

An efficient numerical method for fractional ordinary differential equations based on exponentially decreasing random memory on uniform meshes

On a Coupled System of Fractional Differential Equations with Four Point Integral Boundary Conditions

Contact Info

Product

Resources

About