Abstract
Recently, it was observed that the concentration/heat transfer of pure/nano fluids are finally governed by singular second-order boundary value problems with exponential coefficients. These coefficients were transformed into polynomials and therefore the governing equations become singular in a new independent variable. Unfortunately, the published approximate solutions in the literature suffer from some weaknesses as showed by one of the present coauthors. Hence, the exact solution for such types of problems becomes a challenge. In this paper, a straightforward approach is presented to obtaining the exact solution for such class of singular second-order boundary value problems. The results are also applied to some selected problems within the literature. Accordingly, the published solutions are recovered as special cases of the present ones.
Highlights
The field of boundary value problems frequently arises in many real-life applications.Recently, it was shown that the flow and heat transfer of nanofluids are governed by a system of partial differential equations which are often transformed to a set of ordinary differential equations by using a similarity variable η [1,2,3,4,5,6,7,8]
It was shown that the flow and heat transfer of nanofluids are governed by a system of partial differential equations which are often transformed to a set of ordinary differential equations by using a similarity variable η [1,2,3,4,5,6,7,8]
The given system (7) and (8) reduces to the following homogenous boundary value problem which was analyzed by Ebaid et al [17]: y (t) +
Summary
The field of boundary value problems frequently arises in many real-life applications. It was shown that the flow and heat transfer of nanofluids are governed by a system of partial differential equations which are often transformed to a set of ordinary differential equations by using a similarity variable η [1,2,3,4,5,6,7,8] This set of ordinary differential equations is originally subjected to boundary conditions at infinity. It should be noted that, the boundary conditions at infinity can be transformed to new finite ones by applying a substitution such as t = e− βη , β > 0 [16] With this substitution, the coefficients in the original ordinary differential equations (of exponential orders e− βη , β > 0) are transformed to polynomial ones.
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