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Time-fractional diffusion with mass absorption under harmonic impact

  • Yuriy Povstenko EMAIL logo and Tamara Kyrylych
Published/Copyright: March 13, 2018
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 1

Abstract

Time-fractional diffusion equation with mass absorption and the harmonic source term is studied under zero initial conditions. The Caputo derivative of the order 0 < α ≤ 2 is used. Different formulation of the problem for integer values α = 1 and α = 2 are discussed. The integral transform technique is used. The results of numerical calculations are illustrated graphically.

MSC 2010: Primary 26A33; Secondary 35K05; 45K05
Key Words and Phrases: fractional calculus; Mittag-Leffler functions; fractional partial differential equations; harmonic impact; Fourier transform; Laplace transform

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Appendix A

The inverse Laplace transforms (A.1)–(A.4) are taken from [7], and Eqs. (A.5)–(A.6) from [23]:

L−11s+ps+q=e−qt−e−ptp−q.(A.1)
L−11s2+p2s+q=1p2+q2e−qt−cospt+qpsinpt.(A.2)
L−1exp−qs2+p2s2+p2=J0pt2−q2,t>q>0,0,q>t>0,(A.3)

where J0(r) is the Bessel function of the first kind.

L−1exp−qs2−p2s2−p2=I0pt2−q2,t>q>0,0,q>t>0.(A.4)

Here I0(r) is the modified Bessel function of the first kind.

L−1sμ−νsμ+p=tν−1Eμ,ν−ptμ,(A.5)

where Eμ, ν(z) is the Mittag-Leffler function in two parameters μ and ν:

Eμ,νz=∑k=0∞zkΓμk+ν,μ>0,ν>0,z∈C.(A.6)

Appendix B

The following integrals are taken from [7] and [30]:

∫0∞1x2+p2cosqxdx=π2pe−pq,Rep>0,q>0.(B.1)
∫0∞1x2−p2cosqxdx=−π2psinpq,p>0,q>0.(B.2)

The integral (B.2) is understood in the sense of Cauchy principal value.

∫0∞e−a2x2x2+p2cosqxdx=π4pea2p2[e−pqerfcap−q2a+epqerfcap+q2a],Rea>0,Rep>0,q>0,(B.3)

where erfc (z) is the complementary error function.

∫0∞e−px2cosqxdx=π2pexp−q24p,Rep>0,q>0.(B.4)
∫0∞cosax2+y2x2+p2cosqxdx=π2pe−pqcosay2−p2,p>0,y>0,q>a>0.(B.5)
∫0∞sinax2+y2x2+p2x2+y2cosqxdx=π2pe−pqsinay2−p2y2−p2,p>0,y>0,q>a>0.(B.6)
Received: 2017-10-7
Published Online: 2018-3-13
Published in Print: 2018-2-23

© 2018 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 21–1–2018)
  4. Survey Paper
  5. From continuous time random walks to the generalized diffusion equation
  6. Survey Paper
  7. Properties of the Caputo-Fabrizio fractional derivative and its distributional settings
  8. Research Paper
  9. Exact and numerical solutions of the fractional Sturm–Liouville problem
  10. Research Paper
  11. Some stability properties related to initial time difference for Caputo fractional differential equations
  12. Research Paper
  13. On an eigenvalue problem involving the fractional (s, p)-Laplacian
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  15. Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function
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  19. Optimal control of linear systems with fractional derivatives
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  21. Time-space fractional derivative models for CO2 transport in heterogeneous media
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  23. Improvements in a method for solving fractional integral equations with some links with fractional differential equations
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  25. On some fractional differential inclusions with random parameters
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https://doi.org/10.1515/fca-2018-0008
Keywords for this article
fractional calculus; Mittag-Leffler functions; fractional partial differential equations; harmonic impact; Fourier transform; Laplace transform

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 21–1–2018)
  4. Survey Paper
  5. From continuous time random walks to the generalized diffusion equation
  6. Survey Paper
  7. Properties of the Caputo-Fabrizio fractional derivative and its distributional settings
  8. Research Paper
  9. Exact and numerical solutions of the fractional Sturm–Liouville problem
  10. Research Paper
  11. Some stability properties related to initial time difference for Caputo fractional differential equations
  12. Research Paper
  13. On an eigenvalue problem involving the fractional (s, p)-Laplacian
  14. Research Paper
  15. Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function
  16. Research Paper
  17. Time-fractional diffusion with mass absorption under harmonic impact
  18. Research Paper
  19. Optimal control of linear systems with fractional derivatives
  20. Research Paper
  21. Time-space fractional derivative models for CO2 transport in heterogeneous media
  22. Research Paper
  23. Improvements in a method for solving fractional integral equations with some links with fractional differential equations
  24. Research Paper
  25. On some fractional differential inclusions with random parameters
  26. Research Paper
  27. Initial boundary value problems for a fractional differential equation with hyper-Bessel operator
  28. Research Paper
  29. Mittag-Leffler function and fractional differential equations
  30. Research Paper
  31. Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point
  32. Research Paper
  33. Differential and integral relations in the class of multi-index Mittag-Leffler functions
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