APPLICABLE SOLUTION FOR A CLASS OF ORDINARY DIFFERENTIAL EQUATIONS WITH SINGULARITY

Boundary value problems arise in many real applications such as nanofluids and other areas of applied sciences. The temperature/nanoparticles concentration are usually expressed as singular 2ndorder ODEs. So, it is a challenge to obtain the exact solution of these problems due to the difficulty of the singularity encountered in the governing equations. By means of a suitable transformation, a direct approach is introduced to solve a general class of 2nd-order ODEs. The efficiency of the obtained results is validated through selected problems in the literature. It is found that several existing solutions can be deduced as special cases of our generalized one. Moreover, the present results may be invested for similar future problems in fluid mechanics, especially nanofluids.

Usually, the BCs at infinity are transformed to new finite ones by applying certain substitutions. Accordingly, the coefficients of the ODEs become polynomials. Hence, the temperature/nanoparticles concentration of nanofluids are usually special cases of the following class: under the BCs: where P , Q, l, and R are physical parameters of the nanofluids [1][2][3][4][5][6][7][8]. The constant n takes a particular value according to the final form of the temperature equation, while δ = 0 depends on the final BC. The main objective of this paper is to introduce a direct analysis to exactly solving Eqs. (1.1-1.2). Then, the present generalized results will be invested to construct several exact solutions for some published nanofluids problems as special cases.
In such case, the solution given by Eq. (3.11) becomes we begin with the definition of Kummer's function 1 F 1 (a, b, t): where (a) i is Pochhammer symbol defined as Proofs: From the definition (3.18), 1 F 1 (a, b, −Qt) is given as Since b is neither a negative integer nor zero, then the series (3.20) is defined and its general term v i (x) is given by Implementing the ratio test, we have For finite t and finite Q, we have from (3.22) that such that (3.25) 1 − γ − P > 0, n > −1, (n − γ + 1) a = 0.
Since b = 2 − 2γ − P is neither a negative integer nor zero, then z(t) in (36) is defined. Also, since Q is finite and t is finite in the domain of the problem, t ∈ [0, δ], then 1 F 1 (a, b, −Qt) is convergent by Theorem 1. Also, 1 F 1 (a, b, −Qδ) is convergent because δ is finite and therefore the solution given by Eqs. (3.16-3.17) or its equivalent form (3.24-3.25) converges.
The solution (4.12) can be verified by substitution. It should be mentioned that the solution obtained by Qasim [22] as , does not satisfy Eq. (4.9). Since β > 0 and Sc > 0, then the magnitude Sc β 2 + 1 is never a zero or a negative integer. Hence, the solution (4.12) converges for all positive values of β and Sc.

4.4.
At l = 0, α = 0, δ = Pr β 2 . The following heat transfer equation was also obtained by Qasim [22], where At l = γ1 β 2 , γ is given by which be written as Hence, the solution of the present model is , which is the same exact solution obtained by Qasim [22]. Since β and Pr are always positives, then (2k 1 +1) =   At l = −γ 2 Sc * , we obtain γ as The solution of the present model is in the form: which agrees with Kameswaran et. al [2]. According to the physical values taken by the authors [2], the magnitude d 1 + 1 = (Sc * ) 2 + 4γ 2 Sc * + 1 is always positive and this admits the convergence of the solution (4.26).

Conclusion
In this paper, a general solution was obtained for a class of singular BVPs arise in the field of nanofluids. The solution was derived in terms of the hypergeometric series. The studied class reduced to several published physical models at particular choices of the involved parameters. The obtained solutions were compared with the corresponding results of several models in the literature. It was found that the results in the literature were recovered as special cases of the current ones. Furthermore, this work can be extended in the near future to deal with the recently published physical models [23][24][25].

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.