Approximate Solution of Time-Fractional Helmholtz and Coupled Helmholtz Equations

Mine Aylin Bayrak, Semiha Tetik


The approximate analytical solution of the time-fractional Helmholtz and coupled Helmholtz equations have been acquired successfully via Residual Power Series Method (RPSM). The approximate solutions obtained by RPSM are compared with the exact solutions through different graphics and tables. The fractional derivatives are described in the Caputo sense. The numerical results demonstrate that the new method is quite accurate and readily implemented.


Residual power series method; Time-fractional Helmholtz equation; Caputo derivative; Mittag-Leffler function

Full Text:



O. Abu-Arqub, A. El-Ajou, Z. Al-Zhour and S. Momani, Multiple solutions of nonlinear boundary value problems of fractional order: A new analytic iterative technique, Entropy 16 (2014), 471 – 493, DOI: 10.3390/e16010471.

P. Agarwal and A. A. El-Sayed, Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation, Physica A: Statistical Mechanics and its Applications 500 (2018), 40 – 49, DOI: 10.1016/j.physa.2018.02.014.

R. P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Advances in Difference Equations 2009 (2009), Article number: 981728, URL:

R. L. Bagley and P. J. Torvik, Fractional calculus — A different approach to the analysis of viscoelastically damped structures, AIAA Journal 21 (1983), 741 – 748, DOI: 10.2514/3.8142.

M. A. Bayrak and A. Demir, A new approach for space-time fractional partial differential equations by residual power series method, Applied Mathematics and Computation 336 (2018), 215 – 230, DOI: 10.1016/j.amc.2018.04.032.

M. A. Bayrak, A. Demir and E. Ozbilge, Numerical solution of fractional diffusion equation by Chebyshev collocation method and residual power series method, Alexandria Engineering Journal (2020), DOI: 10.1016/j.aej.2020.08.033.

M. A. Bayrak, A. Demir and E. Ozbilge, On solution of fractional partial differential equation by the weighted fractional operator, Alexandria Engineering Journal (2020), DOI: 10.1016/j.aej.2020.08.044.

S. Das, Solution of extraordinary differential equations with physical reasoning by obtaining modal reaction series, Modelling and Simulation in Engineering 2010 (2010), Article ID 739675, DOI: 10.1155/2010/739675.

A. Demir and M. A. Bayrak, A new approach for the solution of space-time fractional order heat-like partial differential equations by residual power series method, Communications in Mathematics and Applications 10(3) (2019), 585 – 597, DOI: 10.26713/cma.v10i3.626.

A. Demir, M. A. Bayrak and E. Ozbilge, A new approach for the approximate analytical solution of space-time fractional differential equations by the Homotopy Analysis Method, Advances in Mathematical Physics 2009 (2019), Article ID 5602565, DOI: 10.1155/2019/5602565.

E. H. Doha, A. H. Bhrawy, D. Baleanu and S. S. Ezz-Eldien, The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation, Advances in Difference Equations 2014 (2014), Article number: 231, DOI: 10.1186/1687-1847-2014-231.

S. M. El-Sayed and D. Kaya, Comparing numerical methods for Helmholtz equation model problem, Applied Mathematics and Computation 150 (2004), 763 – 773, DOI: 10.1016/S0096-3003(03)00305-9.

A. Ghorbani and A. Alavi, Application of He’s variational iteration method to solve semi differential equations of nth order, Mathematical Problems in Engineering 2008 (2008), Article ID 62798, DOI: 10.1155/2008/627983.

J. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering 167(1-2) (1998), 57 – 68, DOI: 10.1016/S0045-7825(98)00108-X.

K. S. Hedrih, The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam, Mathematical Problems in Engineering 2006 (2006), Article ID 046236, DOI: 10.1155/MPE/2006/46236.

A. A. Hemeda, Homotopy perturbation method for solving partial differential equations of fractional order, International Journal of Mathematical Analysis 6(49-52) (2012), 2431 – 2448, URL:

Y. Hu, Y. Luo and Z. Lu, Analytical solution of the linear fractional differential equation by Adomian decomposition method, Journal of Computational and Applied Mathematics 215 (2008), 220 – 229, DOI: 10.1016/

M. M. Khader, On the numerical solutions for the fractional diffusion equation, Communications in Nonlinear Science and Numerical Simulation 16(6) (2011), 2535 – 2542, DOI: 10.1016/j.cnsns.2010.09.007.

A. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Vol. 204 (2006), 540 pages, Elsevier, URL:

J. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulation 16(3) (2011), 1140 – 1153, DOI: 10.1016/j.cnsns.2010.05.027.

R. Magin, X. Feng and D. Baleanu, Solving fractional order Bloch equation, Concepts in Magnetic Resonance Part A 34A (2009), 16 – 23, DOI: 10.1002/cmr.a.20129.

S. Manabe, A suggestion of fractional-order controller for flexible spacecraft attitude control, Nonlinear Dynamics 29 (2002), 251 – 268, DOI: 10.1023/A:1016566017098.

I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, 368 pages, Academic Press, San Diego (1999), URL:

S. S. Ray and R. K. Bera, Analytical solution of the Bagley Torvik equation by Adomian decomposition method, Applied Mathematics and Computation 168 (2005), 398 – 410, DOI: 10.1016/j.amc.2004.09.006.

A. Sunarto, J. Sulaiman and A. Saudi, Implicit finite difference solution for time-fractional diffusion equations using AOR method, Journal of Physics: Conference Series (2014 International Conference on Science & Engineering in Mathematics, Chemistry and Physics (ScieTech 2014), 13–14 January 2014, Jakarta, Indonesia) 495 (2014), 012032, DOI: 10.1088/1742-6596/495/1/012032.

Y. Tian and A. Chen, The existence of positive solution to three-point singular boundary value problem of fractional differential equation, Abstract and Applied Analysis 2009 (2009), Article ID 314656, DOI: 10.1155/2009/314656.

P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behaviour of real materials, Journal of Applied Mechanics 51 (1984), 294 – 298, DOI: 10.1115/1.3167615.

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, USA (1996), DOI: 10.1007/978-1-4612-4050-1.

Y. Zheng and Z. Zhao, The time discontinuous space-time finite element method for fractional diffusion-wave equation, Applied Numerical Mathematics 150 (2020), 105 – 116, DOI: 10.1016/j.apnum.2019.09.007.



  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905