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Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative

Year 2016, Volume: 4 Issue: 4, 79 - 89, 31.12.2016

Abstract

In this work, we consider a number of boundary-value
problems for time-fractional heat equation with the recently introduced
Caputo-Fabrizio derivative. Using the method of separation of variables, we
prove a unique solvability of the stated problems. Moreover, we have found an
explicit solution to certain initial value problem for Caputo-Fabrizio
fractional order differential equation by reducing the problem to a Volterra
integral equation. Different forms of solution were presented depending on the
values of the parameter appeared in the problem.

References

  • M.Caputo and M.Fabrizio. A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, No 2, 73–85 (2015)
  • M.Caputo and M.Fabrizio. Applications of new time and spatial fractional derivatives with exponential kernels. Progr. Fract. Differ. Appl. 2, No 1, 1–11 (2016)
  • J. Losada and J.J. Nieto. Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, No 2, 87–92 (2015)
  • A.Atangana. On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation. Applied Mathematics and Computation 273 (2016) 948–956.
  • Xiao-Jun Yang, H.M.Srivastava, J.A.Machado Tenreiro. A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, 2015, DOI:10.2298/TSCI151224222Y
  • A.Atangana, D.Baleanu. New fractional derivative with nonlocal and non-singular kernel: Theory and application to hest transfer model. Thermal Science 2016. Online-First Issue 00, Pages 18-18, doi:10.2298/TSC1160111018A(2016)
  • A.Atangana, I.Koca. Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos, Solitons and Fractals, http://dx.doi.org/10.1016/j.chaos.2016.02.012 (2016)
  • V.A. Il’in. Existence of a Reduced System of Eigen- and Associated Functions for a Nonself adjoint Ordinary Differential Operator. Trudy MIAN. 142 (1976) 148–155.
Year 2016, Volume: 4 Issue: 4, 79 - 89, 31.12.2016

Abstract

References

  • M.Caputo and M.Fabrizio. A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, No 2, 73–85 (2015)
  • M.Caputo and M.Fabrizio. Applications of new time and spatial fractional derivatives with exponential kernels. Progr. Fract. Differ. Appl. 2, No 1, 1–11 (2016)
  • J. Losada and J.J. Nieto. Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, No 2, 87–92 (2015)
  • A.Atangana. On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation. Applied Mathematics and Computation 273 (2016) 948–956.
  • Xiao-Jun Yang, H.M.Srivastava, J.A.Machado Tenreiro. A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, 2015, DOI:10.2298/TSCI151224222Y
  • A.Atangana, D.Baleanu. New fractional derivative with nonlocal and non-singular kernel: Theory and application to hest transfer model. Thermal Science 2016. Online-First Issue 00, Pages 18-18, doi:10.2298/TSC1160111018A(2016)
  • A.Atangana, I.Koca. Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos, Solitons and Fractals, http://dx.doi.org/10.1016/j.chaos.2016.02.012 (2016)
  • V.A. Il’in. Existence of a Reduced System of Eigen- and Associated Functions for a Nonself adjoint Ordinary Differential Operator. Trudy MIAN. 142 (1976) 148–155.
There are 8 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Nasser Al-salti This is me

Erkinjon Karimov

Sebti Kerbal This is me

Publication Date December 31, 2016
Published in Issue Year 2016 Volume: 4 Issue: 4

Cite

APA Al-salti, N., Karimov, E., & Kerbal, S. (2016). Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative. New Trends in Mathematical Sciences, 4(4), 79-89.
AMA Al-salti N, Karimov E, Kerbal S. Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative. New Trends in Mathematical Sciences. December 2016;4(4):79-89.
Chicago Al-salti, Nasser, Erkinjon Karimov, and Sebti Kerbal. “Boundary-Value Problems for Fractional Heat Equation Involving Caputo-Fabrizio Derivative”. New Trends in Mathematical Sciences 4, no. 4 (December 2016): 79-89.
EndNote Al-salti N, Karimov E, Kerbal S (December 1, 2016) Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative. New Trends in Mathematical Sciences 4 4 79–89.
IEEE N. Al-salti, E. Karimov, and S. Kerbal, “Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative”, New Trends in Mathematical Sciences, vol. 4, no. 4, pp. 79–89, 2016.
ISNAD Al-salti, Nasser et al. “Boundary-Value Problems for Fractional Heat Equation Involving Caputo-Fabrizio Derivative”. New Trends in Mathematical Sciences 4/4 (December 2016), 79-89.
JAMA Al-salti N, Karimov E, Kerbal S. Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative. New Trends in Mathematical Sciences. 2016;4:79–89.
MLA Al-salti, Nasser et al. “Boundary-Value Problems for Fractional Heat Equation Involving Caputo-Fabrizio Derivative”. New Trends in Mathematical Sciences, vol. 4, no. 4, 2016, pp. 79-89.
Vancouver Al-salti N, Karimov E, Kerbal S. Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative. New Trends in Mathematical Sciences. 2016;4(4):79-8.