Numerical scheme for one-phase 1D fractional Stefan problem using the similarity variable technique

Two fractional Stefan problems are considered by using Riemann-Liouville and Caputo derivatives of order ∈ (0, 1) such that, in the limit case ( = 1), both problems coincide with the same classical Stefan problem. For the one and the other problem, explicit solutions in terms of the Wright functions are presented.We prove that these solutions are different even though they converge, when ↗ 1, to the same classical solution. This result also shows that some limits are not commutative when fractional derivatives are used. KEYWORDSCaputo derivative, explicit solutions, fractional Stefan problem, Riemann-Liouville derivative, Wright functions t 0 t u(x, ) (t − ) d ,6842

show abstract

“…Some works [5][6][7][8][9][10][11] focused in problems like (3). Let us aboard now the physical approach.…”

Section: Introductionmentioning

confidence: 99%

“…we can apply RL 0 D 1− t to both sides of equation (3-i) obtaining, under certain hypothesis, the fractional diffusion equation (6).…”

Section: Introductionmentioning

confidence: 99%

Two different fractional Stefan problems that are convergent to the same classical Stefan problem

Roscani

Tarzia

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Many phenomena in physics, electrical engineering, control theory or mechanics could be better describe using fractional calculus [1,2,3,4,5,10,23,32]. For example, in [1] author used the Atangana-Baleanu derivative with fractional order to further examine chaotic behavior on the Chua's circuit model.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of the Robin boundary condition and order of derivative in time fractional heat conduction equation

Brociek

Słota

2018

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

This paper describes an algorithm for reconstruction the boundary condition and order of derivative for the heat conduction equation of fractional order. This fractional order derivative was applied to time variable and was defined as the Caputo derivative. The heat transfer coefficient, occurring in the boundary condition of the third kind, was reconstructed. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem was solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution an Artificial Bee Colony (ABC) algorithm and Nelder-Mead method were used. In order to stabilize the procedure the Tikhonov regularization was applied. The paper presents examples to illustrate the accuracy and stability of the presented algorithm.

show abstract

“…By using the fractional derivatives, we can describe many types of the phenomena in physics, mechanics, and control theory [4][5][6][7]. For example, the fractional heat conduction equation better describes the distribution of temperature in porous media, than the classical heat conduction equation [8].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of the thermal conductivity coefficient in the space fractional heat conduction equation

Brociek

Słota

2017

THERM SCI

View full text Add to dashboard Cite

In this paper an inverse problem for the space fractional heat conduction equation is investigated. Firstly, we describe the approximate solution of the direct problem. Secondly, for the inverse problem part, we define the functional illustrating the error of approximate solution. To recover the thermal conductivity coefficient we need to minimize this functional. In order to minimize this functional the real ant colony optimization algorithm is used. In the model we apply the Riemann-Liouville fractional derivative. The paper presents also some examples to illustrate the accuracy and stability of the presented algorithm.

show abstract

Numerical scheme for one-phase 1D fractional Stefan problem using the similarity variable technique

Cited by 9 publications

References 8 publications

Two different fractional Stefan problems that are convergent to the same classical Stefan problem

Two different fractional Stefan problems that are convergent to the same classical Stefan problem

Reconstruction of the Robin boundary condition and order of derivative in time fractional heat conduction equation

Reconstruction of the thermal conductivity coefficient in the space fractional heat conduction equation

Contact Info

Product

Resources

About