On multiple solutions for a fourth order nonlinear singular boundary value problems arising in epitaxial growth theory

Abstract

In this article, we consider the fourth order non‐self‐adjoint singular boundary value problem 1rr1rrϕ′′′′=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.

A Study on Solutions for a Class of Higher-Order System of Singular Boundary Value Problem

“…Verma et al [9] analyzed the solutions of Equation ( 6) with a monotone iterative technique and computed the bounds of the growth parameter λ. Recently, they [10] developed the Adomian decomposition method and simulated the solution of (6). All these results and properties make Equation ( 6) very interesting and attractive for researchers.…”

“…To analyze (8), we consider the following three types of homogeneous boundary conditions    lim r→0 r β Θ (r) = 0, Θ (0) = 0, Θ(1) = 0, Θ (1) = 0, lim r→0 r β ϕ (r) = 0, ϕ (0) = 0, ϕ(1) = 0, ϕ (1) = 0, (9) and    lim r→0 r β Θ (r) = 0, Θ (0) = 0, Θ(1) = 0, Θ (1) = 0, lim r→0 r β ϕ (r) = 0, ϕ (0) = 0, ϕ(1) = 0, ϕ (1) = 0, (10) and    lim r→0 r β Θ (r) = 0, Θ (0) = 0, Θ(1) = 0, Θ (1) + Θ (1) = 0, lim r→0 r β ϕ (r) = 0, ϕ (0) = 0, ϕ(1) = 0, ϕ (1) + ϕ (1) = 0. (…”

Two Equivalent Families of Linear Fully Coupled Forward Backward Stochastic Differential Equations

“…In a deterministic LQ problem, it is well known that the control weight in the cost functional must be positive definite, otherwise the optimization problem would not be well-posed (see Anderson and Moore [1], Bensoussan [4]). This kind of linear FBSDEs are widely applied in many areas, such as ordinary differential equations [2,16], stochastic control [5,8,20] and mathematical finance [7,18,22]. However, the well-posedness of ( 4) is not covered by any existing methods despite it is linear, homogeneous and bounded.…”

Existence and nonexistence results for a class of non‐self‐adjoint fourth‐order singular boundary value problems arising in real life

“…They also derived some bounds of the parameter $$$ \lambda $$$ and provide numerical data 11,12 of the solutions. Recently, they 13 applied Adomian decomposition method (ADM) on Equation () for $$$ \alpha =1 $$$.…”