New numerical approach for time-fractional partial differential equations arising in physical system involving natural decomposition method
Saima Rashid, Khadiza Tul Kubra, Asia Rauf, Yu‐Ming Chu, Y. S. Hamed
Abstract
Abstract In the present research, we established an efficient and novel algorithm for time fractional multi-dimensional partial differential equations arising from physics and engineering. Taking into account Caputo fractional derivative, this algorithm involves the fractional natural decomposition method <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="italic">FNDM</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:math> and the nonlinearity term decayed by utilizing the aforesaid method. The solution of the model is based on time dependent fractional-order equations such as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> telegraph equations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="italic">ii</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> parabolic equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="italic">iii</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> Sine-Gordon equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="italic">iv</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> wave-like equations. By considering the FNDM algorithm, the analytic solutions of the aforesaid models are analyzed. The suggested approach employs to solve several models of real-world phenomena and the consequences demonstrate that the approach is reliable, explicit and viable. Moreover, closed form solutions are established in many cases, and exact solutions are derived in particular. Numerical simulations were carried out to ensure that the proposed methods are perfect and precise in terms of efficiency and effectiveness, as shown by the exact solutions resolving complex nonlinear problems. The comparative analysis for the projected method reveals innovative attributes of the hybrid fractional derivative in the discussed model.