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A novel approach for solving linear and nonlinear time-fractional Schrödinger equations

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  • Liaqat, Muhammad Imran
  • Akgül, Ali

Abstract

There is significant literature on Schrödinger differential equation (SDE) solutions, where the fractional derivatives are stated in terms of Caputo derivative (CD). There is hardly any work on analytical and numerical SDE solutions involving conformable fractional derivative (CFD). For the reasons stated above, we are required to solve the SDE in the form of CFD. The main goal of this research is to offer a novel combined computational approach by using conformable natural transform (CNT) and the homotopy perturbation method (HPM) for extracting analytical and numerical solutions of the time-fractional conformable Schrödinger equation (TFCSE) with zero and nonzero trapping potential. We call it the conformable natural transform homotopy perturbation method (CNTHPM). The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the CNTHPM. The error analysis has confirmed the higher degree of accuracy and convergence rates, which indicates the effectiveness and reliability of the suggested method. Furthermore, 2D and 3D graphs compare the exact and approximate solutions. The procedure is quick, precise, and easy to implement, and it yields outstanding results. In addition, numerical results are also compared with other methods such as the differential transform method (DTM), split-step finite difference method (SSFDM), homotopy analysis method (HAM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), and two-dimensional differential transform method (TDDTM). The comparison shows excellent agreement with these methods, which means that CNTHPM is a suitable alternative tool to the methods based on CD for the solutions of the time-fractional SDE. Moreover, we can conclude that the CFD is a suitable alternative to the CD in the modeling of time-fractional SDE. The Banach fixed point theory was also used to test the uniqueness of the solution, convergence, and error analysis.

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  • Liaqat, Muhammad Imran & Akgül, Ali, 2022. "A novel approach for solving linear and nonlinear time-fractional Schrödinger equations," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    • Handle: RePEc:eee:chsofr:v:162:y:2022:i:c:s0960077922006956
    DOI: 10.1016/j.chaos.2022.112487
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    References listed on IDEAS

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    1. Muhammad Imran Liaqat & Adnan Khan & Md. Ashraful Alam & M. K. Pandit & Sina Etemad & Shahram Rezapour & Aida Mustapha, 2022. "Approximate and Closed-Form Solutions of Newell-Whitehead-Segel Equations via Modified Conformable Shehu Transform Decomposition Method," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-14, April.
    2. Naveed Anjum & Chun-Hui He & Ji-Huan He, 2021. "Two-Scale Fractal Theory For The Population Dynamics," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(07), pages 1-10, November.
    3. Ravi Kanth, A.S.V. & Aruna, K., 2009. "Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2277-2281.
    4. Adnan Khan & Muhammad Imran Liaqat & Muhammad Younis & Ashraful Alam & Fairouz Tchier, 2021. "Approximate and Exact Solutions to Fractional Order Cauchy Reaction-Diffusion Equations by New Combine Techniques," Journal of Mathematics, Hindawi, vol. 2021, pages 1-12, December.
    5. Fethi Bin Muhammed Belgacem & Ahmed Abdullatif Karaballi, 2006. "Sumudu transform fundamental properties investigations and applications," International Journal of Stochastic Analysis, Hindawi, vol. 2006, pages 1-23, May.
    6. Shehu Maitama, 2016. "A Hybrid Natural Transform Homotopy Perturbation Method for Solving Fractional Partial Differential Equations," International Journal of Differential Equations, Hindawi, vol. 2016, pages 1-7, September.
    7. Owyed, Saud & Abdou, M.A. & Abdel-Aty, Abdel-Haleem & Alharbi, W. & Nekhili, Ramzi, 2020. "Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    8. Baleanu, Dumitru & Jajarmi, Amin & Mohammadi, Hakimeh & Rezapour, Shahram, 2020. "A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    9. Lokenath Debnath, 2003. "Recent applications of fractional calculus to science and engineering," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-30, January.
    10. Liaqat, Muhammad Imran & Khan, Adnan & Akgül, Ali, 2022. "Adaptation on power series method with conformable operator for solving fractional order systems of nonlinear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    11. Amir Ali & Zamin Gul & Wajahat Ali Khan & Saeed Ahmad & Salman Zeb, 2021. "Investigation Of Fractional Order Sine-Gordon Equation Using Laplace Adomian Decomposition Method," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(05), pages 1-10, August.
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