A highly accurate numerical method for solving nonlinear time‐fractional differential difference equation. (25th August 2019)

• Record Type: Journal Article
• Authors: Khalid, Muhammad; Khan, Fareeha Sami; Sultana, Mariam
• Other Names: Zeidan Dia guestEditor.; Herbst R Sheldon guestEditor.; Lu Tzon‐Tzer guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 10(2021)
• Page Start: 8243
• Page End: 8253
• Publication Date: 2019-08-25
• Subjects: fractional calculus -- nonlinear differential difference equation -- perturbation expansion -- series solution -- Taylor series; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5883
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : This work is based on the implementation of an iterative perturbation method to attain the series solutions of nonlinear fractional differential difference equation (NFDΔE). Perturbation‐iteration algorithm (PIA) assigns a perturbation parameter ϵ to all nonlinear terms and converts it into a simple fractional differential difference equation (FDΔE). By simply solving this FDΔE, series solutions can be obtained. To show the efficacy and accuracy of this method, three famous NFDΔE, i.e, fractional Lotka‐Volterra equation, fractional discrete KdV equation and discretized fractional mKdV lattice equation, will be solved numerically via PIA. Also, comparison of numerical results for α =1 will be done with exact solutions, and their absolute error will also be provided. Graphical illustrations for different values of α will be given to establish the certainty of results. Also, to prove the proficiency among other methods, comparison with different numerical methods is given. Advantage of PIA is that nonlinear terms get vanished during the calculations of Taylor series expansion; therefore, less calculation effort can obtain comprehensive accurate solutions.