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On multiple solutions for a fourth order nonlinear singular boundary value problems arising in epitaxial growth theory
Author(s) -
Verma Amit Kumar,
Pandit Biswajit,
Agarwal Ravi P.
Publication year - 2021
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.7119
Subject(s) - mathematics , uniqueness , boundary value problem , nonlinear system , mathematical analysis , exact solutions in general relativity , singular solution , adomian decomposition method , order (exchange) , differential equation , physics , finance , quantum mechanics , economics
In this article, we consider the fourth order non‐self‐adjoint singular boundary value problem1 rr1 rr ϕ ′′′′ =ϕ ′ϕ ″r + λ , with λ as a parameter measures the speed at which new particles are deposited. This differential equation is non‐self‐adjoint, so finding its solutions is not easy by usual methods. Also, this does not have a unique solution; therefore, finite differences and discrete methods may not be applicable. Here, to find approximate solutions, we propose to use the Adomian decomposition method (ADM) and compute the approximate solutions which are fully dependent on the size of the parameter λ . Here, we prove theoretically that the approximate solutions converge to the exact solutions. For small positive values of parameter λ , the above equation has two solutions. No solutions could be found for large positive values of the parameter. The uniqueness of the solution may not be guaranteed for a fixed positive value of the parameter. For negative values of parameter existences of the solution are always guaranteed.