**5. Solution Technique**

In this present work, the constitutive equations for the given problem are mathematically modeled under the following assumptions, heat generation/absorption, permeable medium, the geometry is taken to be a vertical cone, SWCNTs-water, and MWCNTs-water, convective boundary circumstances. The highly non-linear ODEs were obtained from PDEs by using the transformation technique. Hence, we employed a numerical system on the highly non-linear transformed differential equations.

Equations (12)–(16), along with boundary value problem (17) and (18), called a bvp4c just a name built-in Matlab function which is based on finite difference method and found the solutions computationally which is captured both in tables. Equations (1)–(6) and as well as in graphs. For this purpose, we can transform our differential equations into an arrangement of first order differential equations by letting the new factors

$$f = y\_1, f\nu = y\_2, f\prime = y\_3, \theta = y\_4, \theta\prime = y\_5,\\ S = y\_6, S\prime = y\_7, \gspace{0.1cm} y = y\_8,\\ h = y\_{10}, h\nu = y\_{11} \tag{30}$$

Exercising the above new variables in the Equations (12)–(16) then the following first order differential equations is achieved:

$$\begin{cases} y\_{11} \\ y\_{21} \\ y\_{31} \\ y\_{41} \\ y\_{51} \\ y\_{61} \\ y\_{70} \\ y\_{80} \\ y\_{91} \\ y\_{10} \\ y\_{11} \\ y\_{11} \\ \end{cases} \quad \begin{cases} y\_{2} \\ y\_{3} \\ y\_{2} \\ y\_{3} \\ y\_{4} \\ y\_{5} \\ y\_{6} \\ \end{cases} \quad \begin{cases} y\_{12} \\ y\_{13} \\ y\_{14} \\ y\_{21} \\ y\_{31} \\ y\_{41} \\ y\_{51} \\ \end{cases} \quad \begin{cases} y\_{13} \\ y\_{12} \\ y\_{13} \\ y\_{21} \\ y\_{31} \\ y\_{41} \\ y\_{51} \\ y\_{61} \\ \end{cases} \quad \begin{cases} y\_{14} \\ y\_{15} \\ y\_{16} \\ y\_{21} \\ y\_{31} \\ y\_{41} \\ y\_{51} \\ y\_{61} \\ y\_{70} \\ \end{cases} \quad \begin{cases} y\_{15} \\ y\_{16} \\ y\_{17} \\ y\_{18} \\ y\_{19} \\ y\_{19} \\ y\_{21} \\ y\_{31} \\ y\_{51} \\ y\_{61} \\ \end{cases}$$

With initial conditions

$$\begin{pmatrix} y\_1(0) \\ y\_2(0) \\ y\_2(\infty) \\ y\_2(\infty) \\ y\_5(0) \\ y\_4(\infty) \\ y\_6(0) \\ y\_6(\infty) \\ y\_8(0) \\ y\_9(\infty) \\ y\_{10}(0) \\ y\_{10}(\infty) \end{pmatrix} = \begin{pmatrix} V\_0 \\ 0 \\ 0 \\ -\frac{k\_f}{k\_{nf}}B\_1(1-y\_4(0)) \\ 0 \\ 0 \\ 0 \\ 1-n \\ 0 \\ 1 \\ 0 \end{pmatrix} \tag{32}$$

In this procedure, we can fix the pertinent parameters and then the solution starts with the initial guess supplied at the step size and changes the step size values to get the specified accuracy. The final number of the mesh length is attained by the function of Matlab called bvp4c throughout in the study to get the solutions. The region for the numerical solution should be finite, and the value is taken to be approximately 10 using η = η∞. Since, in the current problem we can find the solution for the SWCNTs-water and MWCNTs-water for a single guess of η∞. Here we have taken the value of η<sup>∞</sup> = <sup>10</sup> for both SWCNT and MWCNT and found the profiles to reach the far field of the boundary layer conditions asymptotically.
