The solutions to the coupled, nonlinear ordinary differential Eqs. (15)–(20) with the boundary conditions (21) are obtained using collocation method with assumed Legendre polynomial. Applying the domain truncation method, the interval \(\left[ {0,\infty } \right)\) is transformed to \(\left[ {0,L} \right]\). The Legendre polynomial is of interval \(\left[ { - 1,1} \right]\) which is transformed to \(\left[ {0,L} \right]\) using the transformation

$$y = \frac{2\eta }{L} - 1.$$

(22)

Hence, the boundary value problem is solved within the region \(\left[ {0,L} \right]\) instead of \(\left[ {0,\infty } \right)\), where \(L\) (scaling parameter) is taken to be sufficiently large enough to take care of the thickness of the boundary layer (Olagunju et al. [20] and Aysun and Salih [21]).

Therefore, the Legendre polynomial is expressed as

$$h_{0} = \sum\limits_{j = 0}^{N} {a_{j} P_{j} \left( y \right)\quad {\text{for}}\,\,j = 0,1,2,3,4, \ldots ,N}$$

(23)

Hence,

$$h_{0} = a_{0} P_{0} \left( y \right) + a_{1} P_{1} \left( y \right) + a_{2} P_{2} \left( y \right) + \cdots$$

(24)

where

$$P_{0} \left( y \right) = 1,\quad P_{1} \left( y \right) = y,\quad P_{2} \left( y \right) = \frac{1}{2}\left( {3y^{2} - 1} \right)$$

(25)

Substituting Eqs. (22) and (25) in Eq. (24) gives

$$h_{0} = a_{0} + a_{1} \left( {2\frac{\eta }{L} - 1} \right) + \frac{{a_{2} }}{2}\left[ {3\left( {2\frac{\eta }{L} - 1} \right)^{2} - 1} \right] + \cdots$$

(26)

For \(L = 20\) and \(N = 20\), Eq. (26) becomes

$$h_{0} = a_{0} + \frac{1}{10}\left( { - 10 + \eta } \right)a_{1} + \frac{1}{200}\left( {200 - 60\eta + 3\eta^{2} } \right)a_{2} + \cdots$$

(27)

Similarly,

$$h_{1} = b_{0} + \frac{1}{10}\left( { - 10 + \eta } \right)b_{1} + \frac{1}{200}\left( {200 - 60\eta + 3\eta^{2} } \right)b_{2} + \cdots$$

(28)

$$\xi_{0} = c_{0} + \frac{1}{10}\left( { - 10 + \eta } \right)c_{1} + \frac{1}{200}\left( {200 - 60\eta + 3\eta^{2} } \right)c_{2} + \cdots$$

(29)

$$\xi_{1} = d_{0} + \frac{1}{10}\left( { - 10 + \eta } \right)d_{1} + \frac{1}{200}\left( {200 - 60\eta + 3\eta^{2} } \right)d_{2} + \cdots$$

(30)

$$\zeta_{0} = e_{0} + \frac{1}{10}\left( { - 10 + \eta } \right)e_{1} + \frac{1}{200}\left( {200 - 60\eta + 3\eta^{2} } \right)e_{2} + \cdots$$

(31)

$$\zeta_{1} = f_{0} + \frac{1}{10}\left( { - 10 + \eta } \right)f_{1} + \frac{1}{200}\left( {200 - 60\eta + 3\eta^{2} } \right)f_{2} + \cdots$$

(32)

\(\left\{ {a_{0} , a_{1} , \ldots , a_{N} } \right\},\) \(\left\{ {b_{0} , b_{1} , \ldots , b_{N} } \right\},\) \(\left\{ {c_{0} , c_{1} , \ldots , c_{N} } \right\},\) \(\left\{ {d_{0} , d_{1} , \ldots , d_{N} } \right\},\) \(\left\{ {e_{0} , e, \ldots ., e_{N} } \right\}\) and \(\left\{ {f_{0} , f, \ldots , f_{N} } \right\}\) are unknown coefficients which can be obtained using the coupled, nonlinear differential Eqs. (15)–(20) with the boundary conditions (21). Hence, the approximate solutions of the truncated series (27)–(32) can be obtained.

$${\text{Collpoints}} = {\text{NSolve}}\left[ {{\text{Expand}}\left[ {{\text{LegendreP}}\left[ {N - 1, 2\frac{\eta }{L} - 1} \right]} \right] = 0,\eta } \right]$$

(33)

Equation (33) is a MATHEMATICA Software Language used to generate the following collocation points for values of \(\eta\).

$$\left. {\begin{array}{l} {0.0759316,\quad 0.397918,\quad 0.968441,\quad 1.77285,\quad 2.79034,\quad 3.99455,} \\ {5.35429,\quad 6.83436,\quad 8.39641,\quad 10.0000,\quad 11.6036,\quad 13.1657,} \\ {14.646,\quad 16.0051,\quad 17.2097,\quad 18.2273,\quad 19.0319,\quad 19.6002,\quad 19.9241} \\ \end{array} } \right\}$$

(34)

Equations (27)–(32) are substituted in Eqs. (15)–(20), with the default values for the fluid parameters; \(H_{{\text{g}}} = 5\), \(M_{{\text{g}}} = 5\), \(P_{{\text{r}}} = 0.71\), \(E_{{\text{c}}} = 0.01\), \(\psi = \frac{\pi }{2}\), \(k_{1} = 0.1\), \(S_{{\text{c}}} = 0.22\), \(R_{{\text{d}}} = 1.6\), \(K = 0.1\), \(D_{{\text{u}}} = 0.1\), \(S_{{\text{t}}} = 0.2\), \(M = 1\), \(t = 0.1\), \(A = 0.5\), \(S = 0.2\), \(\lambda = 0.5\), \(\omega = 1\), and \(\varepsilon = 0.1\), to give six residual equations. By imposing the boundary conditions (21) on Eqs. (27)–(32), twelve equations are derived. Each residual equation is collocated at the above collocation points to yield one hundred and fourteen collocation equations. Consequently, there are a total number of one hundred and twenty-six equations with one hundred and twenty-six unknown coefficients. These equations are solved using MATHEMATICA 11.0 software. The numerical values obtained for the unknown coefficients are then substituted back into Eqs. (27)–(32). Hence, Eq. (14) becomes

$$\begin{aligned} h & = 0.631525 + 5.71453 \times 10^{ - 25} i - \left( {0.0248978 + 3.12151 \times 10^{ - 25} i} \right) \left( { - 10 + \eta } \right) \\ & \quad + \left( {0.000790613 + 7.21393 \times 10^{ - 27} i} \right)\left( {200 - 60\eta + 3\eta^{2} } \right) \\ & \quad - \left( {0.0000480709 - 8.45787 \times 10^{ - 27} i} \right)\left( { - 400 + 240\eta - 30\eta^{2} + \eta^{3} } \right) - \\ \end{aligned}$$

(35)

$$\begin{aligned} & \xi = 0.140729 + 1.49935 \times 10^{ - 24} i - {(}0.030269 + 7.79491 \times 10^{ - 25} i) - 10 + \eta \\ & \quad + (0.00136063 + 1.65512 \times 10^{ - 26} i)(200 - 60\eta + 3\eta^{2} ) \\ & \quad - (0.0041331 - 2.16109 \times 10^{ - 26} i)( - 400 + 240\eta - 30\eta^{2} + \eta^{3} ) + \ldots \\ \end{aligned}$$

(36)

$$\begin{aligned} \zeta & = 0.{194627} - 2.1262 \times 10^{ - 25} i - (00373656 - 8.6365 \times 10^{ - 26} i)( - 10 + \eta ) \\ & \quad + (0.00134708 - 1.58859 \times 10^{ - 27} i)(200 - 60\eta + 3\eta^{2} ) \\ & \quad (0.000299884 + 2.32537 \times 10^{ - 27} i)( - 400 + 240\eta - 30\eta^{2} + \eta^{3} ) + \ldots \\ \end{aligned}$$

(37)

Skin-friction:

The coefficient of skin-friction in non-dimensional form is expressed as

$$C_{f} = \left( {\frac{\partial }{\partial \eta }\left( {h_{0} \left( \eta \right) + \varepsilon e^{i\omega t} h_{1} \left( \eta \right)} \right)} \right)_{\eta = 0}$$

(38)

Nusselt Number

The rate of heat transfer at the plate is expressed as

$${\text{Nu}} = - \left( {\frac{\partial }{\partial \eta }\left( {\xi_{0} \left( \eta \right) + \varepsilon e^{i\omega t} \xi_{1} \left( \eta \right)} \right)} \right)_{\eta = 0}$$

(39)

Sherwood Number

The rate of mass transfer in non-dimensional form is given as

$${\text{Sh}} = - \left( {\frac{\partial }{\partial \eta }\left( {\zeta_{0} \left( \eta \right) + \varepsilon e^{i\omega t} \zeta_{1} \left( \eta \right)} \right)} \right)_{\eta = 0}$$

(40)