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

Abstract

In this article, we propose a fourth-order non-self-adjoint system of singular boundary value problems (SBVPs), which arise in the theory of epitaxial growth by considering hte equation 1rβrβ1rβ(rβΘ′)′′′=12rβK11μ′Θ′2+2μΘ′Θ″+K12μ′φ′2+2μφ′φ″+λ1G1(r),1rβrβ1rβ(rβφ′)′′′=12rβK21μ′Θ′2+2μΘ′Θ″+K22μ′φ′2+2μφ′φ″+λ2G2(r), where λ1≥0 and λ2≥0 are two parameters, μ=pr2β−2,p∈R+, G1,G2∈L1[0,1] such that M1*≥G1(r)≥M1>0,M2*≥G2(r)≥M2>0 and K12>0, K11≥0, and K21>0, K22≥0 are constants that are connected by the relation (K12+K22)≥(K11+K21) and β>1. To study the governing equation, we consider three different types of homogeneous boundary conditions. We use the transformation t=r1+β1+β to deduce the second-order singular boundary value problem. Also, for β=p=G1(r)=G2(r)=1, it admits dual solutions. We show the existence of at least one solution in continuous space. We derive a sign of solutions. Furthermore, we compute the approximate bound of the parameters to point out the region of nonexistence. We also conclude bounds are symmetric with respect to two different transformations.