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Some analytical and numerical investigation of a family of fractional‐order Helmholtz equations in two space dimensions
Author(s) -
Srivastava Hari M.,
Shah Rasool,
Khan Hassan,
Arif Muhammad
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5846
Subject(s) - mathematics , helmholtz free energy , helmholtz equation , fractional calculus , laplace transform , mathematical analysis , adomian decomposition method , space (punctuation) , partial differential equation , order (exchange) , calculus (dental) , computer science , boundary value problem , medicine , physics , dentistry , finance , quantum mechanics , economics , operating system
In this article, we aim at solving a family of two‐dimensional fractional‐order Helmholtz equations by using the Laplace‐Adomian Decomposition Method (LADM). The fractional‐order derivatives, which we use in this investigation, follows the Liouville‐Caputo definition. Our results based upon the LADM are obtained in series form that helps us in analyzing the analytical solutions of the fractional‐order Helmholtz equations considered here. For illustration and verification of the analytical procedure using the LADM, several numerical examples and graphical representations are presented for the analytical solution of the fractional‐order Helmholtz equations. The mathematical analytic procedure, which we have used here, has shown that the LADM is a fairly accurate and computable method for the solution of problems involving fractional‐order Helmholtz equations in two dimensions. In an analogous manner, one can apply the LADM for finding the analytical solution of other classes of fractional‐order partial differential equations.