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A semi-analytic method for fractional-order ordinary differential equations: Testing results

  • Sergiy Reutskiy EMAIL logo and Zhuo-Jia Fu
Published/Copyright: February 9, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 6

Abstract

The paper presents the testing results of a semi-analytic collocation method, using five benchmark problems published in a paper by Xue and Bai in Fract. Calc. Appl. Anal., Vol. 20, No 5 (2017), pp. 1305–1312, DOI: 10.1515/fca-2017-0068.

MSC 2010: Primary 26A33; Secondary 65M70
Key Words and Phrases: fractional differential equation; semi-analytic method; initial problem; benchmark problems

5 Appendix

  1. Let us find the Caputo derivative Dtα [Ψm], 0 < α ≤ 1 of the discontinuous function

    Ψmt=0,0≤t≤tmt−tmp,tm<t,DtαΨmt=def1Γ1−α∫0tΨm1τt−ταdτ.

    Using

    Ψm1t=0,0≤t≤tmpt−tmp−1,tm<t,

    we get:

    DtαΨmt=def0,0≤t≤tm<2,pΓ1−α∫tmtτ−tmp−1t−ταdτ,tm<t≤2,pΓ1−α∫tmtτ−tmp−1t−ταdτ using τ=ζ+tm=pΓ1−α∫0t−tmζ+tm−tmp−1t−ζ−tmαdζ=pΓ1−α∫0t−tmζp−1t−tm−ζαdζ=t−tm=spΓ1−α∫0sζp−1s−ζαdζ=pΓ1−α∫0sζp−1s−ζ1−α−1dζ=pΓ1−αΓpΓ1−αΓp+1−αsp−α=pΓpΓp+1−αsp−α=Γp+1Γp+1−αsp−α=Γp+1Γp+1−αt−tmp−α.

    Here we use the Euler integral:

    ∫0tζp−1t−ζq−1dζ=ΓpΓqΓp+qtp+q−1.

    Finally, we obtain:

    DtαΨmt=0,0≤t≤tm<2,Γp+1Γp+1−αt−tmp−α,tm<t≤2.
  2. Dt2ut=AtutDt0.5ut−utt2+e−2t.

    Let us consider u,v=Dt0.5ut, and w = ut(t) as independent variables,

    Dt2u=Atuv−w2+e−2t

    Suppose that u0, v0, w0 are given functions of t which are the initial approximations of the corresponding exact values. Then we have the following relations:

    u=u0+(u−u0)=u0+δu,v=v0+(v−v0)=v0+δv,w=w0+(w−w0)=w0+δw.

    Assuming that δu, δv, δw are small values, the equation can be linearized by the following way:

    uv=u0+δuv0+δv≃v0δu+u0δv+u0v0=v0(u−u0)+u0(v−v0)+u0v0=v0u+u0v−u0v0,w2=w0+δw2≃w02+2w0δw=w02+2w0(w−w0)=2w0w−w02.

    As a result, we get the linear equation with respect to u, v, w:

    Dt2u=Atv0u+u0v−u0v0−2w0w−w02+e−2t,Dt2u=Atv0tu+Atu0tv−2w0tw+e−2t−Atu0tv0t−w02t.

    Returning to the old notations, we get the linear equation

    Dt2u=Atv0tu+Atu0tDt0.5u−2w0tu1+e−2t−Atu0tv0t−w02t,

    or

    Dtνu≡Dt2u=Btu+CtDt0.5u+Etu1+Ft⇒⇒Dtνu=Btu+CtDtν1u+Etu1+Ft,

    where

    Bt=Atv0t=AtDt0.5u0,Ct=Atu0t,Et=−2w0t=−2u01t,Ft=e−2t−Atu0tDt0.5u0−u01t2.

  3. Using the quasilinearization procedure, we can write:

    wsvs≃w0sv0s+sw0s−1v0sw−w0+sw0sv0s−1v−v0=sw0s−1v0sw+sw0sv0s−1v+1−2sw0sv0s.

    So, the right hand side of the equation (65) can be written in the form:

    Avsws+fIt≃Asw0s−1v0sw+Asw0sv0s−1v+A1−2sw0sv0s+fIt=Ptw+Rtv+Ft,

    where

    Pt=Asw0s−1v0s,Rt=Asw0sv0s−1,Ft=A1−2sw0sv0s+fIt.

Acknowledgements

The work described in this paper was supported by the National Science Fund of China (Grant Nos. 11772119,11572111), the Foundation for Open Project of State Key Laboratory of Structural Analysis for Industrial Equipment (Grant No. GZ1707), the Sino-Ukraine Science and Technology Cooperation Project (Grant No. CU02-18), the Fundamental Research Funds for the Central Universities (Grant No. 2016B06214), Alexander von Humboldt Research Fellowship (ID: 1195938) and Qing Lan Project.

References

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Received: 2017-12-26
Published Online: 2019-02-09
Published in Print: 2018-12-19

© 2018 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books
  4. Research Paper
  5. Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions
  6. Finite-time attractivity for semilinear tempered fractional wave equations
  7. Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas
  8. Extrapolating for attaining high precision solutions for fractional partial differential equations
  9. Time optimal controls for fractional differential systems with Riemann-Liouville derivatives
  10. Inverses of generators of integrated fractional resolvent operator functions
  11. A variational approach for boundary value problems for impulsive fractional differential equations
  12. Infinitely many solutions to boundary value problem for fractional differential equations
  13. A semi-analytic method for fractional-order ordinary differential equations: Testing results
  14. Blow-up and global existence of solutions for a time fractional diffusion equation
  15. A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity
  16. Short Paper
  17. Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions
https://doi.org/10.1515/fca-2018-0084
Keywords for this article
fractional differential equation; semi-analytic method; initial problem; benchmark problems

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books
  4. Research Paper
  5. Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions
  6. Finite-time attractivity for semilinear tempered fractional wave equations
  7. Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas
  8. Extrapolating for attaining high precision solutions for fractional partial differential equations
  9. Time optimal controls for fractional differential systems with Riemann-Liouville derivatives
  10. Inverses of generators of integrated fractional resolvent operator functions
  11. A variational approach for boundary value problems for impulsive fractional differential equations
  12. Infinitely many solutions to boundary value problem for fractional differential equations
  13. A semi-analytic method for fractional-order ordinary differential equations: Testing results
  14. Blow-up and global existence of solutions for a time fractional diffusion equation
  15. A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity
  16. Short Paper
  17. Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions
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