SOLVING NONLINEAR TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING THE RESIDUAL POWER SERIES METHOD
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Date
2026
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Abstract
This memoir deals with the solution of nonlinear fractional-time partial differen
tial equations using the Residual Power Series Method (RPSM). The method is
applied to the generalized Kuramoto–Sivashinsky (KS) equation, adopting the Ca
puto time-fractional derivative of order α. The memoir is divided into three chap
ters: the first chapter covers the fundamentals of fractional calculus and special
functions, the second reviews the RPSM methodology and its convergence theory,
while the third presents a numerical application to three test cases with different pa
rameters and initial conditions. The resulting approximate solutions are compared
with exact solutions and the results of other numerical methods through error ta
bles and two- and three-dimensional graphical plots. The results demonstrate high
accuracy, numerical stability, and low computational cost. They also highlight the
memory effect induced by the fractional order α on wave behavior and temporal
convergence, confirming the efficiency of the method in modeling complex phys
ical phenomena such as flame fronts, thin liquid films, and plasma.