Approximate solutions to nonlinear fractional order partial differential equations arising in ion-acoustic waves

Samia Bushnaq; Sajjad Ali; Kamal Shah; Muhammad Arif; Samia Bushnaq; Sajjad Ali; Kamal Shah; Muhammad Arif

doi:10.3934/math.2019.3.721

AIMS Mathematics

2019, Volume 4, Issue 3: 721-739. doi: 10.3934/math.2019.3.721

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Approximate solutions to nonlinear fractional order partial differential equations arising in ion-acoustic waves

1.
Department of Basic Sciences, Princess Sumaya University for Technology, Amman 11941, Jordan
2.
Department of Mathematics, Shaheed Benazir Bhutto University Sheringal Dir(U), Khyber Pakhtunkhwa, Pakistan
3.
Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan
4.
Department of Mathematics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan

Received: 17 March 2019 Accepted: 02 June 2019 Published: 21 June 2019
MSC : 35A22, 35A25, 35K57

An approximative procedure named asymptotic homotopy perturbation method (AHPM) is introduced to obtain solutions of the non-linear fractional order models. The two special cases, FZK$(3, 3, 3)$ and FZK$(2, 2, 2)$ of fractional Zakharov-Kuznetsov equations are chosen for the illustrative purpose of our method. AHPM is a very recent new procedure as compare with other existing homotopy perturbation procedures. A new auxiliary function has been introduced in AHPM. The AHPM solutions are compared with solutions of fractional complex transform FCT using variational iteration method VIM and exact solutions. Further, the surface graph of AHPM solutions are compared with surface graph of solutions of homotopy perturbation transform method (HPTM) solutions. In comparison, the solutions computed by AHPM are in agreement with exact solutions of the problems. The simulation section reveals that our new developed procedure is effective and explicit.
- asymptotic homotopy perturbation method,
- fractional Zakharov-Kuznetsov equations
Citation: Samia Bushnaq, Sajjad Ali, Kamal Shah, Muhammad Arif. Approximate solutions to nonlinear fractional order partial differential equations arising in ion-acoustic waves[J]. AIMS Mathematics, 2019, 4(3): 721-739. doi: 10.3934/math.2019.3.721

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Abstract

An approximative procedure named asymptotic homotopy perturbation method (AHPM) is introduced to obtain solutions of the non-linear fractional order models. The two special cases, FZK$(3, 3, 3)$ and FZK$(2, 2, 2)$ of fractional Zakharov-Kuznetsov equations are chosen for the illustrative purpose of our method. AHPM is a very recent new procedure as compare with other existing homotopy perturbation procedures. A new auxiliary function has been introduced in AHPM. The AHPM solutions are compared with solutions of fractional complex transform FCT using variational iteration method VIM and exact solutions. Further, the surface graph of AHPM solutions are compared with surface graph of solutions of homotopy perturbation transform method (HPTM) solutions. In comparison, the solutions computed by AHPM are in agreement with exact solutions of the problems. The simulation section reveals that our new developed procedure is effective and explicit.

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