Cross diffusion effects on MHD double diffusive viscous flow through Hermite wavelet method

Abstract

Double-diffusive convection is a form of fluid flow that occurs when two processes of molecular diffusion are active in a fluid at the same time, causing instabilities and also complicated behaviour. One chemical or biological species concentration can cause a flux of another species, either linearly or nonlinearly, a phenomenon known as cross-diffusion. The cross-diffusion effects on double-diffusive MHD fluid flow through the Hermite wavelet method is examined. The governing coupled partial differential equations of the problem under consideration are transformed to highly nonlinear ordinary differential equations over a finite domain with the help of similarity transformations. The results are obtained for the skin friction coefficient, as well as the velocity, temperature and the concentration profiles for some values of the governing parameters, namely, the cross diffusion terms, Hartmann number, thermophoresis parameter, squeeze number, Prandtl number and suction/injection parameter. The obtained results are validated against previously published results for special case of the problems.

1 Introduction

Heat and mass transfer are two important concepts in the fields of thermodynamics, fluid mechanics, and engineering [1, 2]. They describe the transfer of energy and the transfer of matter from one place to another. Thermal energy moving from a region with a higher temperature to one with a lower temperature is referred to as heat transfer. Mass transfer is the process of the movement of a substance from one place to another place due to differences in concentration. In the context of mass transfer, convection refers to the bulk movement of fluid that carries a substance with it. This is common in applications such as chemical reactors. In some cases, mass transfer can also occur within solid materials, such as in the diffusion of atoms through a crystal lattice. These processes are fundamental in various fields, including chemical engineering, mechanical engineering, and environmental science. Natural convection heat and mass transfer in partially heated vertical parallel plates are explored by Lee [3]. An analytical results for fully urbanized thermohaline mixed convection between parallel vertical plates was studied by Boulama and Galanis [4]. Yan et al. [5] are investigated double diffusive convection between vertical parallel plates with asymmetric heating. Miyamoto [6] is studied heat transfer through between vertical parallel plates in an anisotropic permeable layer with throughflow. Capone [7] is deliberated thermohaline penetrative convection through internal heating.

The terms suction and injection can have various implications relying upon the setting where they are utilized [8,9,10]. In general, suction refers to the process of drawing or pulling something, typically a fluid or gas, into a vacuum or low-pressure area. This is often done using a pump, vacuum cleaner, or similar device. In the medical field, suction may refer to the removal of fluids, such as mucus or blood, from a patient’s body using a suction device. In plumbing, suction can refer to the drawing of water or other liquids through a pipe or hose using a pump or siphon. Injecting a drug or vaccine into a patient's body is referred to as an injection in medical contexts. In the manufacturing context, injecting molten plastic material into a mould cavity is a common process known as injection forming, which produces plastic parts. When discussing programming and software, the term injection can be used to describe a number of methods, such as SQL injection, which involves inserting malicious code into a database or program. Without more specific context, it’s challenging to provide a more detailed explanation. In a similar flow between parallel plates, Hamza [11] is studied the effects of injection and suction on a similar flow between parallel plates. A heated vertical plate that was moving continuously was used for the mixed convection heat transfer, which Al-Sanea [12] arranged using either injection or suction.

Cross-diffusion refers to the occurrence when the gradient of concentration or density of one biological or chemical species triggers the movement of another species, either in a linear or nonlinear manner. Models involving cross-diffusion consist of a set of interconnected parabolic equations that depict the changes in densities or concentrations within a system comprising multiple components. These models find widespread utility across various scientific disciplines such as physics, chemistry, and biology. Lou and Ni [13] are studied diffusion vs cross-diffusion an elliptic approach compared the two phenomena. In their study, Iida [14] looked into cross-diffusion, diffusion, and competitive interaction. Using analysis and simulations, Madzvamuse [15] examined cross-diffusion-driven reaction instability systems. Ruiz-Baier [16] investigated effects of cross-diffusion through mathematical analysis and numerical simulation. Almirantis [17] investigated how cross-diffusion affects the formation of chemical and biological patterns. Amplitude equations for cross-diffusion reaction–diffusion systems were studied by Zemskov [18]. Malashetty and Gaikwad [19] investigated the effect of cross-diffusion on thermohaline convection. Bothe [20] concentrated on multicomponent diffusion using the Maxwell–Stefan method.

Magnetohydrodynamics (MHD) is a multidisciplinary field of science and engineering that combines principles from both magnetism and fluid dynamics. It focuses on the behaviour of fluids that conduct electricity, including liquid metals, plasmas, and some electrolytes, when they are exposed to magnetic fields. MHD has applications [21,22,23] in nuclear fusion research, space propulsion, metallurgy, and energy production. In nuclear fusion research, MHD plays a critical role in replicating the energy production processes of stars on Earth. Magnetic confinement techniques like tokamaks and stellarators use MHD principles to control and stabilize the high-temperature plasma needed for fusion. Some advanced propulsion concepts, such as magneto plasma or ion propulsion, use MHD to accelerate ionized propellant through the interaction of magnetic and electric fields. In industrial applications, MHD has been explored for enhancing heat transfer and fluid flow in liquid metal cooling systems used in some nuclear reactors. A collection of basic formulas that characterize the behavior of electrically conducting fluids in the occurrence of magnetic fields govern MHD. The foundation of these formulas is the conservation of mass, momentum, energy, and magnetic induction. The complex, nonlinear nature of MHD makes it a challenging field. Researchers and engineers work to understand and control phenomena such as magnetic instabilities, turbulence, and the interaction of magnetic fields with the flow of conducting fluids. MHD is at the intersection of plasma physics, fluid dynamics, and electromagnetism and continues to be an area of active research and development, especially in the context of clean energy production and space exploration. Salah [24] utilized a finite element method for magnetohydrodynamics. In Font [25], general relativity's numerical hydrodynamics and MHD were utilized. Numerical MHD algorithm and tests for one-dimensional flow were applied by Ryu and Jones [26] in astrophysics. The numerical techniques for multi-physical MHD were examined by Bég [27]. A numerical method for general relativistic MHD was studied by De Villiers and Hawley [28]. Meng [29] extended anisotropic ion pressure in classical and semi relativistic MHD. MHD in small fluidic channels with spatially irregular magnetic fields was studied by Das [30] a framework for the combined MHD and magnetophoretic particle transport. Maintaining pressure positivity in MHD simulations was investigated by Balsara and Spicer [31]. Lohrasbi and Sahai [32] are closely examined MHD heat transfer in a two-phase flow between parallel plates. MHD nanofluid flow under the influence of viscous dissipation is numerically examined by Gopal et al. [33]. A study by Seth et al. [34] explored thermohaline and thermo-diffusion effects on MHD non-Newtonian fluid flow in a non-Darcy permeable layer with Newtonian heating. Amouzadeh et al. [35] are discussed the impact of suction/injection on MHD liquid flow inside a vertical annulus. The numerical solution of MHD channel flow in the occurrence of an inclined magnetic field in a permeable layer with uniform suction and injection was studied by Chutia [36]. Homotopy perturbation solution for the analysis of MHD suction or injection squeeze flow between two parallel disks was obtained by Domairry and Aziz [37]. Upreti et al. [38] obtained MHD flow of Ag-water nanofluid over a flat porous plate with viscous-Ohmic dissipation, suction/injection and heat generation/absorption.

Overall, the study of double-diffusive MHD fluid flow with Dufour and Soret effects is a complex and challenging problem in fluid mechanics. It is very difficult to solve highly non-linear fluid flow problems by using analytically. Hence, the analysis of such flows requires the use of advanced mathematical and numerical techniques, and it can provide valuable insights into the behaviour of fluids under different thermal and concentration gradients. There are various types of numerical methods available in the literature, and we consider one such wavelet numerical method. The study of wavelets is a relatively new and developing field in mathematics. It has found applications in various technical fields. However, wavelets have proven to be highly effective in signal analysis, especially in representing and segmenting waveforms, performing time–frequency analysis, and implementing quick algorithms in a straightforward manner [39]. Wavelets allow a wide range of functions and operators to be accurately represented [40, 41] and also provide a relationship with quick numerical methods [42]. Many researchers contributions towards wavelet based numerical methods are as follows: Laguerre wavelets method for Lane–Emden equation [43], Laguerre wavelets collocation method [44], cardinal B-spline [45], Bernoulli wavelet [46], Hermite wavelet method [47], new generalized Hermite wavelet method [48], etc. Therefore, developing a new kind of nonlinear numerical Hermite wavelet method that requires no perturbation parameter is necessary. Nevertheless, with the development of parametric computational software including MATHEMATICA, MAPLE, MATLAB, etc., approximate numerical Hermite wavelet method for nonlinear challenges have been approved by various researchers. Recently, some interesting boundary layer fluid flow problems [49,50,51,52,53,54,55] are solved by using numerical Hermite wavelet method. Recently, some boundary layer fluid flow problems are solved by using different numerical methods [56,57,58,59].

To the best of our knowledge, MHD double diffusive viscous fluid flow with cross diffusion terms through HWM have not received any attention in the literature. The intent of the present study is to solve highly non-linear coupled differential equations with the help of HWM. The implementation of this method was thoroughly discussed in our earlier work [49,50,51,52,53,54,55]. The impact of the physical parameters such as the Prandtl number, Squeeze number, Schmidt number Hartmann number, suction/blowing parameter, cross diffusion parameters on velocity and temperature distributions in the boundary layer are examined. The association of this article is as follows; Sect. 2 is devoted to the Mathematical framework of the problem. The basics of Hermite wavelet method is available in Sect. 3 and also methodology of HWM is given in Sect. 4. The results are discussed in Sect. 5 and finally, the conclusion is drawn in Sect. 6.

2 Mathematical framework

Double diffusive fluid flow between two infinite parallel plate with distance $h\left( t \right) = H\sqrt {1 - \alpha t}$ and a uniform magnetic field of strength $B\left( t \right) = \left( {0,0,\frac{B_0 }{{\sqrt {1 - \alpha t} }}} \right)$ is applied perpendicular to the disks with suction/injection. Approaching the stationary lower disk at $z = 0$, the upper disk at $z = h\left( t \right)$ is moving at a velocity of $\frac{ - \alpha H}{{2\sqrt {1 - \alpha t} }}$. The cylindrical coordinate system $\left( {r,\theta ,z} \right)$ is considered. Rotational symmetry was assumed for the flow, with the velocity being defined as $\vec{q} = \left( {{\text{u}}\left( {{\text{r }},{\text{ z}}} \right),0,{\text{w}}\left( {{\text{r }},{\text{ z}}} \right)} \right)$, where u and w represent the axial and radial velocities, respectively, along with the $r$ and $z$ axes. The governing equations are expressed as follows [60]

$$\frac{\partial u}{{\partial r}} + \frac{\partial w}{{\partial z}} = - \frac{u}{r},$$

(1)

$$\frac{\partial u}{{\partial t}} + u\frac{\partial u}{{\partial r}} + w\frac{\partial u}{{\partial z}} = - \frac{1}{\rho }\frac{\partial p}{{\partial r}} + \frac{\mu }{\rho }\left( {\frac{\partial^2 u}{{\partial r^2 }} + \frac{\partial^2 u}{{\partial z^2 }} + \frac{1}{r}\frac{\partial u}{{\partial r}} - \frac{1}{r^2 }u} \right) - \frac{\sigma B^2 }{\rho }u,$$

(2)

$$\frac{\partial w}{{\partial t}} + u\frac{\partial w}{{\partial r}} + w\frac{\partial w}{{\partial z}} = - \frac{1}{\rho }\frac{\partial p}{{\partial z}} + \frac{\mu }{\rho }\left( {\frac{\partial^2 w}{{\partial r^2 }} + \frac{\partial^2 w}{{\partial z^2 }} + \frac{1}{r}\frac{\partial w}{{\partial r}}} \right),$$

(3)

$$\frac{\partial T}{{\partial t}} + u\frac{\partial T}{{\partial r}} + w\frac{\partial T}{{\partial z}} = D_{11} \left( {\frac{\partial^2 T}{{\partial r^2 }} + \frac{\partial^2 T}{{\partial z^2 }} + \frac{1}{r}\frac{\partial T}{{\partial r}}} \right) + D_{12} \left( {\frac{\partial^2 C}{{\partial r^2 }} + \frac{\partial^2 C}{{\partial z^2 }} + \frac{1}{r}\frac{\partial C}{{\partial r}}} \right),$$

(4)

$$\frac{\partial C}{{\partial t}} + u\frac{\partial C}{{\partial r}} + w\frac{\partial C}{{\partial z}} = D_{21} \left( {\frac{\partial^2 T}{{\partial r^2 }} + \frac{\partial^2 T}{{\partial z^2 }} + \frac{1}{r}\frac{\partial T}{{\partial r}}} \right) + D_{22} \left( {\frac{\partial^2 C}{{\partial r^2 }} + \frac{\partial^2 C}{{\partial z^2 }} + \frac{1}{r}\frac{\partial C}{{\partial r}}} \right).$$

(5)

where $u$ and $w$ axial and radial velocities along with $z$ and $r$ axes, $C$ is concentration, $T$ is the temperature,$\rho$ is fluid density,$\sigma$ is electrical conductivity, $p$ is the pressure,$B$ is the magnetic field,$\mu$ is the dynamic viscosity,$\alpha$ is a characteristic parameter having dimensions of period inverse,$D_{11}$ is the thermal diffusivity,$D_{12}$ and $D_{21}$ are the cross diffusion diffusivities, and $D_{22}$ is the solute an analogous of $D_{11} .$

The boundary conditions of the problem are [60]

$$T = T_H ,w = \frac{dh}{{dt}},\;u = 0\;{\text{at}}\;z = h\left( t \right)$$

(6)

$$T = T_w ,\;w = \frac{ - w_0 }{{\sqrt {1 - \alpha t} }},\;u = 0\,{\text{at}}\;z = 0$$

We introduce following similarity transformations

$$\eta = \frac{z}{{H\left( {1 - \alpha t} \right)}},\;u = \frac{\alpha r}{{2\left( {1 - \alpha t} \right)}}f^{\prime}\left( \eta \right),w = \frac{ - \alpha H}{{\left( {1 - \alpha t} \right)^\frac{1}{2} }}f\left( \eta \right),\; \theta = \frac{T - T_H }{{T_W - T_H }},\;\phi = \frac{C - C_H }{{C_W - C_H }}$$

(7)

where $\eta$ is the dimensionless co-ordinate in the z‐direction,$\theta$ is the temperature function,$\phi$ is concentration function, $T_H$ is the surface temperature at the upper disk, $T_W$ is the surface temperature at the lower disk,$t$ is time, $\frac{d}{dt}$ is the material derivative, $C_H$ is surface concentration at the upper disk, $C_w$ is surface concentration at the lower disk,$w_0$ is the initial velocity and $H$ is hall current.

By substituting the above similarity variables into Eqs. (1) to (6), and removing the pressure gradient from the resulting equations gives

$$f^{iv} - S\left( {3f^{\prime \prime} + \eta f^{{\prime} {\prime} {\prime} } - 2ff^{{\prime} {\prime} {\prime} }} \right) - M^2 f^{\prime \prime} = 0,$$

(8)

$$\theta^{\prime \prime} + Pr S\left( {f\theta^{\prime} - \eta \theta^{\prime}} \right) + \gamma_{12} \phi^{\prime \prime} = 0,$$

(9)

$$\phi^{\prime \prime} + Sc S\left( {f\phi^{\prime} - \eta \phi^{\prime}} \right) + \gamma_{21} \theta^{\prime \prime} = 0.$$

(10)

The corresponding boundary conditions are

$$f(0) = A,f^{\prime}(1) = f^{\prime}(0) = 0,f(1) = 0.5,\;\;\theta \left( 0 \right) = 1, \phi \left( 0 \right) = 1, \phi \left( 1 \right) = 0 = \theta \left( 1 \right)$$

(11)

where $S = \frac{\alpha H^2 }{{2\nu_f }}$ is the squeeze number, $M = \frac{\sigma B_0^2 H^2 }{\mu }$ is the Hartmann number, $\Pr = \frac{\nu_f }{{D_{11} }}$ is the Prandtl number, $A = \frac{w_0 }{{\alpha H}}$ is the suction or blowing parameter,$Sc = \frac{\nu_f }{{D_{22} }}$ is the Schmidt number, $\gamma_{12} = \frac{C_H }{{T_H }}\frac{{D_{12} }}{{D_{11} }}$ is the Dufour quantity, $\gamma_{21} = \frac{T_H }{{C_H }}\frac{{D_{21} }}{{D_{22} }}$ is the Soret quantity.

3 Background of Hermite wavelet method

The Hermite wavelet is one of the continuous polynomial basis wavelet is studied [47,48,49]. Now, We approximate the solution $y(x)$ of the nonlinear differential equation under Hermite wavelet space is as follows:

$$y(x) = \sum_{n = 1}^\infty {\sum_{m = 0}^\infty {C_{n,m} \,\,\psi_{n,m} (x),} }$$

where $\psi_{n,m} (x)$ is given in [47,48,49, 51,52,53,54]. We approximate $y(x)$ by truncating the series as follows

$$y(x) \approx \sum_{n = 1}^{2^{k - 1} } {\sum_{m = 0}^{M - 1} {C_{n,m\,} \psi_{n,m} (x)} = A^T \psi (x),}$$

where A and $\psi (x)$ are $2^{k - 1} M \times 1$ matrix,

$$A^T = [C_{1,0} , \ldots C_{1,M - 1} ,C_{2,0} \ldots C_{2,M - 1} , \ldots C_{2^{k - 1} ,0} , \ldots C_{2^{k - 1} ,M - 1} ],$$

$$\psi (x) = [\psi_{1,0} , \ldots \psi_{1,M - 1} ,\psi_{2,0} , \ldots \psi_{2,M - 1} , \ldots \psi_{2^{k - 1} ,0} , \ldots \psi_{2^{k - 1} ,M - 1} ]^T .$$

3.1 Operational matrix of integration

Here, we extracted some Hermite wavelet basis at k = 1 as follows:

$$\begin{gathered} \psi_{1,0} (x) = \frac{2}{{\sqrt {\pi } }} \hfill \\ \psi_{1,1} (x) = \frac{1}{{\sqrt {\pi } }}(8x - 4) \hfill \\ \psi_{1,2} (x) = \frac{1}{{\sqrt {\pi } }}(32x^2 - 32x + 4) \hfill \\ \psi_{1,3} (x) = \frac{1}{{\sqrt {\pi } }}(128x^3 - 192x^2 + 48x + 8) \hfill \\ \psi_{1,4} (x) = \frac{1}{{\sqrt {\pi } }}(512x^4 - 1024x^3 + 384x^2 + 128x - 40) \hfill \\ \end{gathered}$$

$$\psi_{1,5} (x) = \frac{1}{{\sqrt {\pi } }}(2048x^5 - 5120x^4 + 2560x^3 + 1280x^2 - 800x + 16)$$

$$\psi_{1,6} (x) = \frac{1}{{\sqrt {\pi } }}(8192x^6 - 24576x^5 + 15360x^4 + 10240x^3 - 9600x^2 + 384x + 368)$$

$$\psi_{1,7} (x) = \frac{1}{{\sqrt {\pi } }}(32768x^7 - 114688x^6 + 86016x^5 + 71680x^4 - 89600x^3 + 5376x^2 + 10304x - 928)$$

$$\begin{gathered} \psi_{1,8} (x) = \frac{1}{{\sqrt {\pi } }}(131072x^8 - 524288x^7 + 458752x^6 + 458752x^5 - 716800x^4 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+ 57344x^3 + 164864x^2 - 29696x - 3296) \hfill \\ \end{gathered}$$

$$\begin{gathered} \psi_{1,9} (x) = \frac{1}{{\sqrt {\pi } }}(524288x^9 - 2359296x^8 + 2359296x^7 + 2752512x^6 - 5160960x^5 + 516096x^4 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +1978368x^3 - 534528x^2 - 118656x + 21440) \hfill \\ \end{gathered}$$

$$\begin{gathered} \psi_{1,10} (x) = \frac{1}{{\sqrt {\pi } }}(2097152x^{10} - 10485760x^9 + 11796480x^8 + 15728640x^7 - 34406400x^6 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +4128768x^5 + 19783680x^4 - 7127040x^3 - 2373120x^2 + 857600x + 16448) \hfill \\ \end{gathered}$$

$$\begin{gathered} \psi_{1,11} (x) = \frac{1}{{\sqrt {\pi } }}(8388608x^{11} - 46137344x^{10} + 57671680x^9 + 86507520x^8 - 216268800x^7 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +30277632x^6 + 174096384x^5 - 78397440x^4 - 34805760x^3 + 18867200x^2 + 723712x - 461696) \hfill \\ \end{gathered}$$

$$\begin{gathered} \psi_{1,12} (x) = \frac{1}{{\sqrt {\pi } }}(33554432x^{12} - 201326592x^{11} + 276824064x^{10} + 461373440x^9 - 1297612800x^8 + 207618048x^7 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +1392771072x^6 - 752615424x^5 - 417669120x^4 + 301875200x^3 + 17369088x^2 - 22161408x + 561536) \hfill \\ \end{gathered}$$

where $\psi_9 (x) = [\psi_{1,0} (x),\psi_{1,1} (x),\psi_{1,2} (x),\,\psi_{1,3} (x),\,\psi_{1,4} (x),\,\psi_{1,5} (x),\,\psi_{1,6} (x),\psi_{1,7} (x),\psi_{1,8} (x)]^T$.

Now integrate above first nine basis concerning x limit from 0 to x, then express as a linear combination of Hermite wavelet basis as,

$$\int_0^x {\psi_{1,0} (x)} = \left[ {\begin{array}{*{20}c} \frac{1}{2} & \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\psi_{1,1} (x)} = \left[ {\begin{array}{*{20}c} {\frac{ - 1}{4}} & 0 & \frac{1}{8} & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\psi_{1,2} (x)} = \left[ {\begin{array}{*{20}c} {\frac{ - 1}{3}} & 0 & 0 & \frac{1}{12} & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\psi_{1,3} (x)} = \left[ {\begin{array}{*{20}c} \frac{5}{4} & 0 & 0 & 0 & \frac{1}{16} & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\psi_{1,4} (x)} = \left[ {\begin{array}{*{20}c} {\frac{ - 2}{5}} & 0 & 0 & 0 & 0 & \frac{1}{20} & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\psi_{1,5} (x)} = \left[ {\begin{array}{*{20}c} {\frac{ - 23}{3}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{24} & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\psi_{1,6} (x)} = \left[ {\begin{array}{*{20}c} \frac{116}{7} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{28} & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\psi_{1,7} (x)} = \left[ {\begin{array}{*{20}c} \frac{103}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{32} \\ \end{array} } \right]\,\psi_9 (x)$$

$\int_0^x {\psi_{1,8} (x)} = \left[ {\begin{array}{*{20}c} {\frac{ - 2680}{9}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x) + \frac{1}{36}\psi_{1,9} (x)$.

Hence,

$$\int_0^x {\,\psi (x)} \,dx = H_{9 \times 9} \,\,\psi_9 (x) + \,\,\overline{\psi }_9 (x)$$

where,

$$H_{9 \times 9} = \left[ {\begin{array}{*{20}c} {\tfrac{1}{2}} & {\tfrac{1}{4}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{ - 1}{4}} & 0 & {\tfrac{1}{8}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{ - 1}{3}} & 0 & 0 & {\tfrac{1}{12}} & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{5}{4}} & 0 & 0 & 0 & {\tfrac{1}{16}} & 0 & 0 & 0 & 0 \\ {\tfrac{ - 2}{5}} & 0 & 0 & 0 & 0 & {\tfrac{1}{20}} & 0 & 0 & 0 \\ {\tfrac{ - 23}{3}} & 0 & 0 & 0 & 0 & 0 & {\tfrac{1}{24}} & 0 & 0 \\ {\tfrac{116}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & {\tfrac{1}{28}} & 0 \\ {\tfrac{103}{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\tfrac{1}{32}} \\ {\tfrac{ - 2680}{9}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],\,\,\overline{\psi_9 }(x) = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ {\frac{1}{36}\psi_{1,9} (x)} \\ \end{array} } \right].$$

Next, twice integration of above nine basis is given below

$$\int_0^x {\int_0^x {\psi_{1,0} (x)} \,dxdx} = \left[ {\begin{array}{*{20}c} \frac{3}{16} & \frac{1}{8} & \frac{1}{32} & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\psi_{1,1} (x)} \,dxdx} = \left[ {\begin{array}{*{20}c} {\frac{ - 1}{6}} & {\frac{ - 1}{{16}}} & 0 & \frac{1}{96} & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\psi_{1,2} (x)} \,dxdx} = \left[ {\begin{array}{*{20}c} {\frac{ - 1}{{16}}} & {\frac{ - 1}{{12}}} & 0 & 0 & \frac{1}{192} & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\psi_{1,3} (x)} \,dxdx} = \left[ {\begin{array}{*{20}c} \frac{3}{5} & \frac{5}{16} & 0 & 0 & 0 & \frac{1}{320} & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\psi_{1,4} (x)} \,dxdx} = \left[ {\begin{array}{*{20}c} {\frac{ - 7}{{12}}} & {\frac{ - 1}{{10}}} & 0 & 0 & 0 & 0 & \frac{1}{480} & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\psi_{1,5} (x)} \,dxdx} = \left[ {\begin{array}{*{20}c} {\frac{ - 22}{7}} & {\frac{ - 23}{{12}}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{672} & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\psi_{1,6} (x)} \,dxdx} = \left[ {\begin{array}{*{20}c} \frac{81}{8} & \frac{29}{7} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{896} \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\psi_{1,7} (x)} \,dxdx} = \left[ {\begin{array}{*{20}c} \frac{148}{9} & \frac{103}{8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x) + \frac{1}{1152}\psi_{1,9} (x)$$

$$\int_0^x {\int_0^x {\psi_{1,8} (x)} \,dxdx} = \left[ {\begin{array}{*{20}c} {\frac{ - 773}{5}} & {\frac{ - 670}{9}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x) + \frac{1}{1440}\psi_{1,10} (x).$$

Hence,

$$\int_0^x {\int_0^x {\,\psi (x)} \,dx} dx = H^{\prime}_{9 \times 9} \,\,\psi_9 (x) + \,\,\overline{{\psi^{\prime}}}_9 (x)$$

where,

$$H^{\prime}_{9 \times 9} = \left[ {\begin{array}{*{20}c} {\tfrac{3}{16}} & {\tfrac{1}{8}} & {\tfrac{1}{32}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{ - 1}{6}} & {\tfrac{ - 1}{{16}}} & 0 & {\tfrac{1}{96}} & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{ - 1}{{16}}} & {\tfrac{ - 1}{{12}}} & 0 & 0 & {\tfrac{1}{192}} & 0 & 0 & 0 & 0 \\ {\tfrac{3}{5}} & {\tfrac{5}{16}} & 0 & 0 & 0 & {\tfrac{1}{320}} & 0 & 0 & 0 \\ {\tfrac{ - 7}{{12}}} & {\tfrac{ - 1}{{10}}} & 0 & 0 & 0 & 0 & {\tfrac{1}{480}} & 0 & 0 \\ {\tfrac{ - 22}{7}} & {\tfrac{ - 23}{{12}}} & 0 & 0 & 0 & 0 & 0 & {\tfrac{1}{672}} & 0 \\ {\tfrac{81}{8}} & {\tfrac{29}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & {\tfrac{1}{896}} \\ {\tfrac{148}{9}} & {\tfrac{103}{8}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{ - 773}{5}} & {\tfrac{ - 670}{9}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],\,\,\overline{{\psi^{\prime}_9 }}(x) = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ {\frac{1}{1152}\psi_{1,9} (x)} \\ {\frac{1}{1440}\psi_{1,10} (x)} \\ \end{array} } \right].$$

Again triple integration of above nine basis is given by,

$$\int_0^x {\int_0^x {\int_0^x {\psi_{1,0} (x)} \,dxdx} dx} = \left[ {\begin{array}{*{20}c} \frac{5}{96} & \frac{3}{64} & \frac{1}{64} & \frac{1}{384} & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\psi_{1,1} (x)} \,dxdx} dx} = \left[ {\begin{array}{*{20}c} {\frac{ - 7}{{128}}} & {\frac{ - 1}{{24}}} & {\frac{ - 1}{{128}}} & 0 & \frac{1}{1536} & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\psi_{1,2} (x)} \,dxdx} dx} = \left[ {\begin{array}{*{20}c} {\frac{ - 1}{{80}}} & {\frac{ - 1}{{64}}} & {\frac{ - 1}{{96}}} & 0 & 0 & \frac{1}{3840} & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\psi_{1,3} (x)} \,dxdx} dx} = \left[ {\begin{array}{*{20}c} {\frac{19}{{96}}} & \frac{3}{20} & \frac{5}{128} & 0 & 0 & 0 & \frac{1}{7680} & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\psi_{1,4} (x)} \,dxdx} dx} = \left[ {\begin{array}{*{20}c} {\frac{ - 13}{{56}}} & {\frac{ - 7}{{48}}} & {\frac{ - 1}{{80}}} & 0 & 0 & 0 & 0 & \frac{1}{13440} & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\psi_{1,5} (x)} \,dxdx} dx} = \left[ {\begin{array}{*{20}c} {\frac{ - 65}{{64}}} & {\frac{ - 11}{{14}}} & {\frac{ - 23}{{96}}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{21504} \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\,\psi_{1,6} (x)} \,dxdx} dx} = \left[ {\begin{array}{*{20}c} {\frac{133}{{36}}} & {\frac{81}{{32}}} & {\frac{29}{{56}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x) + \frac{1}{32256}\psi_{1,9} (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\psi_{1,7} (x)} \,dxdx} dx} = \left[ {\begin{array}{*{20}c} {\frac{193}{{40}}} & \frac{37}{9} & {\frac{103}{{64}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x) + \frac{1}{46080}\psi_{1,10} (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\psi_{1,8} (x)} \,dxdx} dx} = \left[ {\begin{array}{*{20}c} {\frac{ - 1211}{{22}}} & {\frac{ - 773}{{20}}} & {\frac{ - 335}{{36}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x) + \frac{1}{63360}\psi_{1,11} (x).$$

Hence,

$$\int_0^x {\int_0^x {\int_0^x {\,\psi (x)} \,dx} dxdx} = H^{\prime \prime}_{9 \times 9} \,\,\psi_9 (x) + \,\,\overline{{\psi^{\prime \prime}_9 }}(x)$$

where,

$$H^{\prime \prime}_{9 \times 9} = \left[ {\begin{array}{*{20}c} {\tfrac{5}{96}} & {\tfrac{3}{64}} & {\tfrac{1}{64}} & {\tfrac{1}{384}} & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{ - 7}{{128}}} & {\tfrac{ - 1}{{24}}} & {\tfrac{ - 1}{{128}}} & 0 & {\tfrac{1}{1536}} & 0 & 0 & 0 & 0 \\ {\tfrac{ - 1}{{80}}} & {\tfrac{ - 1}{{64}}} & {\tfrac{ - 1}{{96}}} & 0 & 0 & {\tfrac{1}{3840}} & 0 & 0 & 0 \\ {\tfrac{19}{{96}}} & {\tfrac{3}{20}} & {\tfrac{5}{128}} & 0 & 0 & 0 & {\tfrac{1}{7680}} & 0 & 0 \\ {\tfrac{ - 13}{{56}}} & {\tfrac{ - 7}{{48}}} & {\tfrac{ - 1}{{80}}} & 0 & 0 & 0 & 0 & {\tfrac{1}{13440}} & 0 \\ {\tfrac{ - 65}{{64}}} & {\tfrac{ - 11}{{14}}} & {\tfrac{ - 23}{{96}}} & 0 & 0 & 0 & 0 & 0 & {\tfrac{1}{21504}} \\ {\tfrac{133}{{36}}} & {\tfrac{81}{{32}}} & {\tfrac{29}{{56}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{193}{{40}}} & {\tfrac{37}{9}} & {\tfrac{103}{{64}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{ - 1211}{{22}}} & {\tfrac{ - 773}{{20}}} & {\tfrac{ - 335}{{36}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],\overline{{\psi^{\prime \prime}_9 }}(x) = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ {\frac{1}{32256}\psi_{1,9} (x)} \\ {\frac{1}{46080}\psi_{1,10} (x)} \\ {\frac{1}{63360}\psi_{1,11} (x)} \\ \end{array} } \right].$$

Again fourth integration of the above nine basis is given by,

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\psi_{1,0} (x)} \,dxdx} dxdx} } = \left[ {\begin{array}{*{20}c} {\frac{19}{{1536}}} & \frac{5}{384} & \frac{3}{512} & \frac{1}{768} & \frac{1}{6144} & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\psi_{1,1} (x)} \,dxdx} dxdx} } = \left[ {\begin{array}{*{20}c} {\frac{ - 7}{{480}}} & {\frac{ - 7}{{512}}} & {\frac{ - 1}{{192}}} & {\frac{ - 1}{{1536}}} & 0 & \frac{1}{30720} & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\psi_{1,2} (x)} \,dxdx} dxdx} } = \left[ {\begin{array}{*{20}c} {\frac{ - 1}{{1152}}} & {\frac{ - 1}{{320}}} & {\frac{ - 1}{{512}}} & {\frac{ - 1}{{1152}}} & 0 & 0 & \frac{1}{92160} & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\psi_{1,3} (x)} \,dxdx} dxdx} } = \left[ {\begin{array}{*{20}c} {\frac{17}{{336}}} & {\frac{19}{{384}}} & \frac{3}{160} & \frac{5}{1536} & 0 & 0 & 0 & \frac{1}{215040} & 0 \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\psi_{1,4} (x)} \,dxdx} dxdx} } = \left[ {\begin{array}{*{20}c} {\frac{ - 55}{{768}}} & {\frac{ - 13}{{224}}} & {\frac{ - 7}{{384}}} & {\frac{ - 1}{{960}}} & 0 & 0 & 0 & 0 & \frac{1}{430080} \\ \end{array} } \right]\,\psi_9 (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\psi_{1,5} (x)} \,dxdx} dxdx} } = \left[ {\begin{array}{*{20}c} {\frac{{ - {53}}}{{{216}}}} & {\frac{ - 65}{{256}}} & {\frac{ - 11}{{112}}} & {\frac{ - 23}{{1152}}} & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\psi_9 (x) + \frac{1}{774144}\psi_{1,9} (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\psi_{1,6} (x)} \,dxdx} dxdx} } = \left[ {\begin{array}{*{20}c} {\frac{497}{{480}}} & {\frac{133}{{144}}} & {\frac{81}{{256}}} & {\frac{29}{{672}}} & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x) + \frac{1}{1290240}\psi_{1,10} (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\psi_{1,7} (x)} \,dxdx} dxdx} } = \left[ {\begin{array}{*{20}c} {\frac{127}{{132}}} & {\frac{193}{{160}}} & {\frac{37}{{72}}} & {\frac{103}{{768}}} & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x) + \frac{{1}}{{{2027520}}}\psi_{1,11} (x)$$

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\psi_{1,8} (x)} \,dxdx} dxdx} } = \left[ {\begin{array}{*{20}c} {\frac{ - 4277}{{288}}} & {\frac{ - 1211}{{88}}} & {\frac{ - 773}{{160}}} & {\frac{ - 335}{{432}}} & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\,\psi_9 (x) + \frac{1}{{{3041280}}}\psi_{1,12} (x).$$

Hence,

$$\int_0^x {\int_0^x {\int_0^x {\int_0^x {\,\psi (x)} \,dx} dxdx} dx = H^{{\prime} {\prime} {\prime} }_{9 \times 9} \,\,\psi_9 (x) + \,\,\overline{{\psi^{{\prime} {\prime} {\prime} }_9 }}(x)}$$

where

$$H_{9 \times 9}^{{\prime} {\prime} {\prime} } = \left[ {\begin{array}{*{20}c} {\tfrac{{19}}{{1536}}} & {\tfrac{5}{{384}}} & {\tfrac{3}{{512}}} & {\tfrac{1}{{768}}} & {\tfrac{1}{{6144}}} & 0 & 0 & 0 & 0 \\ {\tfrac{{ - 7}}{{480}}} & {\tfrac{{ - 7}}{{512}}} & {\tfrac{{ - 1}}{{192}}} & {\tfrac{{ - 1}}{{1536}}} & 0 & {\tfrac{1}{{30720}}} & 0 & 0 & 0 \\ {\tfrac{{ - 1}}{{1152}}} & {\tfrac{{ - 1}}{{320}}} & {\tfrac{{ - 1}}{{512}}} & {\tfrac{{ - 1}}{{1152}}} & 0 & 0 & {\tfrac{1}{{92160}}} & 0 & 0 \\ {\tfrac{{17}}{{336}}} & {\tfrac{{19}}{{384}}} & {\tfrac{3}{{160}}} & {\tfrac{5}{{1536}}} & 0 & 0 & 0 & {\tfrac{1}{{215040}}} & 0 \\ {\tfrac{{ - 55}}{{768}}} & {\tfrac{{ - 13}}{{224}}} & {\tfrac{{ - 7}}{{384}}} & {\tfrac{{ - 1}}{{960}}} & 0 & 0 & 0 & 0 & {\tfrac{1}{{430080}}} \\ {\tfrac{{ - 53}}{{216}}} & {\tfrac{{ - 65}}{{256}}} & {\tfrac{{ - 11}}{{112}}} & {\tfrac{{ - 23}}{{1152}}} & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{{497}}{{480}}} & {\tfrac{{133}}{{144}}} & {\tfrac{{81}}{{256}}} & {\tfrac{{29}}{{672}}} & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{{127}}{{132}}} & {\tfrac{{193}}{{160}}} & {\tfrac{{37}}{{72}}} & {\tfrac{{103}}{{768}}} & 0 & 0 & 0 & 0 & 0 \\ {\tfrac{{ - 4277}}{{288}}} & {\tfrac{{ - 1211}}{{88}}} & {\tfrac{{ - 773}}{{160}}} & {\tfrac{{ - 335}}{{432}}} & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],\overline{{\psi _9^{{\prime} {\prime} {\prime} } }}(x) = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ {\frac{1}{{774144}}\psi _{1,9} (x)} \\ {\frac{1}{{1290240}}\psi _{1,10} (x)} \\ {\frac{1}{{2027520}}\psi _{1,11} (x)} \\ {\frac{1}{{3041280}}\psi _{1,12} (x)} \\ \end{array} } \right].$$

In the same way, we can create matrices for our convenience. In the result section, the fourth-order nonlinear differential equation is solved by taking N=9. So, we generated matrices of order 9*9 up to the fourth-order operational matrix of integration.

4 Hermite wavelet numerical methodology

4.1 Velocity profile $f$:

Let us consider

$$f^{iv} (x) = H^T \psi \left( x \right).$$

(12)

Integrating above equation w. r. t $x$ and the range from $0$ to $x$

$$f^{{\prime\prime\prime} } (x) - f^{{\prime\prime\prime} } (0) = H^T \left[ {\psi \left( x \right)G + \overline{\psi }(x)} \right].$$

(13)

Integrating above equation w. r. t $x$ and the range from $0$ to $x$

$$f^{\prime\prime} (x) - f^{\prime\prime} (0) - xf^{{\prime} {\prime} {\prime} } (0) = H^T \left[ {\psi \left( x \right)G^{\prime} + \overline{{\psi^{\prime} }}(x)} \right].$$

(14)

Integrating above equation w. r. t $x$ and the range from $0$ to $x$

$$f^{\prime}(x) - f^{\prime}(0) = xf^{\prime \prime}(0) + \frac{x^2 }{2}f^{{\prime} {\prime} {\prime} }(0) + H^T \left[ {\psi \left( x \right)G^{\prime \prime} + \overline{{\psi^{\prime \prime}}}(x)} \right].$$

(15)

Integrating above equation w. r. t $x$ and the range from $0$ to $x$ and put $f^{\prime}(0) = 0,\,\,\,\,f(0) = A$, we have

$$f(x) = A + \frac{{x^2 }}{2}f^{\prime\prime} (0) + \frac{{x^3 }}{6}f^{{\prime} {\prime} {\prime} } (0) + H^T \left[ {\psi \left( x \right)G^{{\prime} {\prime} {\prime} } + \overline{{\psi ^{{\prime} {\prime} {\prime} } }}(x)} \right].$$

(16)

Substitute $f^{\prime}(1) = 0$$,f(1) = 0.5$ in Eqs. (15) and (16) and solve those equations to get $f^{{\prime} {\prime} {\prime} } (0)$ and $f^{\prime \prime}(0),$ we have.

$$f^{{\prime} {\prime} {\prime} }(0) = 12A - 6 + 12H^T \left. {\left[ {\psi \left( x \right)G^{{\prime} {\prime} {\prime} } + \overline{{\psi^{{\prime} {\prime} {\prime} }}}(x)} \right]} \right|_{x = 1} - 6H^T \left. {\left[ {\psi \left( x \right)G^{\prime \prime} + \overline{{\psi^{\prime \prime}}}(x)} \right]} \right|_{x = 1} ,$$

(17)

$$\begin{aligned} f^{\prime \prime}(0) = & - \frac{1}{2}\left( {12A - 6 + 12H^T \left. {\left[ {\psi \left( x \right)G^{{\prime} {\prime} {\prime} } + \overline{{\psi^{{\prime} {\prime} {\prime} }}}(x)} \right]} \right|_{x = 1} - 6H^T \left. {\left[ {\psi \left( x \right)G^{\prime \prime} + \overline{{\psi^{\prime \prime}}}(x)} \right]} \right|_{x = 1} } \right) \\ & - H^T \left. {\left[ {\psi \left( x \right)G^{\prime \prime} + \overline{{\psi^{\prime \prime}}}(x)} \right]} \right|_{x = 1} . \\ \end{aligned}$$

(18)

Substitute (17) and (18) in (16) we get

$$\begin{aligned} f(x) = & A + \frac{x^2 }{2}\left( \begin{gathered} - \frac{1}{2}\left( {12A - 6 + 12H^T \left. {\left[ {\psi \left( x \right)G^{{\prime} {\prime} {\prime} } + \overline{\psi }(x)} \right]} \right|_{x = 1} - 6H^T \left. {\left[ {\psi \left( x \right)G^{\prime \prime} + \overline{{\psi^{\prime \prime}}}(x)} \right]} \right|_{x = 1} } \right) \hfill \\ - H^T \left. {\left[ {\psi \left( x \right)G^{\prime \prime} + \overline{{\psi^{\prime \prime}}}(x)} \right]} \right|_{x = 1} \hfill \\ \end{gathered} \right) \\ & + \frac{x^3 }{6}\left( {12A - 6 + 12H^T \left. {\left[ {\psi \left( x \right)G^{{\prime} {\prime} {\prime} } + \overline{{\psi^{{\prime} {\prime} {\prime} }}}(x)} \right]} \right|_{x = 1} - 6H^T \left. {\left[ {\psi \left( x \right)G^{\prime \prime} + \overline{{\psi^{\prime \prime}}}(x)} \right]} \right|_{x = 1} } \right). \\ & + H^T \left[ {\psi \left( x \right)G^{{\prime} {\prime} {\prime} } + \overline{{\psi^{{\prime} {\prime} {\prime} }}}(x)} \right] \\ \end{aligned}$$

(19)

4.2 Temperature profile $\left( \theta \right)$

$$\theta^{\prime \prime}(x) = H^T \psi (x)$$

(20)

Integrating above equation w. r. t $x$ and the range from $0$ to $x$

$$\theta^{\prime}(x) = \theta^{\prime}(0) + H^T [\psi (x)G + \overline{\psi }(x)]$$

(21)

Integrating above equation w. r. t $x$ and the range from $0$ to $x$ and put $\theta (0) = 1$

$$\theta (x) = 1 + x\theta^{\prime}(0) + H^T [\psi (x)G^{\prime} + \overline{{\psi^{\prime}}}(x)]$$

(22)

Substitute $\theta \left(1\right)=0$, we have

$$\theta^{\prime}(0) = - 1 - H^T \left. {[\psi (x)G^{\prime} + \overline{{\psi^{\prime}}}(x)]} \right|_{x = 1}$$

(23)

Substitute Eq. (23) in Eq. (22), we get.

$$\theta (x) = 1 + x\left( { - 1 - H^T \left. {[\psi (x)G^{\prime} + \overline{{\psi^{\prime}}}(x)]} \right|_{x = 1} } \right) + H^T [\psi (x)G^{\prime} + \overline{{\psi^{\prime}}}(x)].$$

(24)

4.3 Concentration profile $(\phi )$

$$\phi^{\prime \prime}(x) = V^T \psi (x).$$

(25)

Integrating above equation w. r. t $x$ and the range from $0$ to $x$

$$\phi^{\prime}(x) = \phi^{\prime}(0) + V^T [\psi (x)G + \overline{\psi }(x)].$$

(26)

Integrating above equation w. r. t $x$ and the range from $0$ to $x$ and put $\phi (0) = 1$.

$$\phi (x) = 1 + x\phi^{\prime}(0) + V^T [\psi (x)G^{\prime} + \overline{{\psi^{\prime}}}(x)].$$

(27)

Substitute $\phi (1) = 0$, we have.

$$\phi^{\prime}(0) + 1 = - V^T \left. {[\psi (x)G^{\prime} + \overline{{\psi^{\prime}}}(x)]} \right|_{x = 1} .$$

(28)

Substitute Eq. (28) in Eq. (27), we get.

$$\phi (x) = 1 + x\left( { - 1 - V^T \left. {[\psi (x)G^{\prime} + \psi^{\prime}(x)]} \right|_{x = 1} } \right) + V^T [\psi (x)G^{\prime} + \overline{{\psi^{\prime}}}(x)]$$

(29)

Substitute $f^{{\prime} {\prime} {\prime} {\prime} }$$f^{{\prime} {\prime} {\prime} },$$f^{\prime \prime},$$f^{\prime},$$f$, $\theta^{\prime \prime},$ $\theta^{\prime},$ $\theta ,$ and $\phi^{\prime \prime},$ $\phi^{\prime},$ $\phi$ in the governing equations, then collate these equations using the collection steps as${x}_{i}=\frac{2i-1}{2N}$,, where$i=\mathrm{1,2},3,\dots N$. We have solved a nonlinear systems of algebraic equations by using Newton's Raphson method with the help of MATHEMATICA. We obtain values of the Hermite wavelets unknown coefficients. Substitute these values in Eqs. (19, 24, 29) gives numerical solutions for Eqs. (8–11).

5 Results and discussions

The governing partial differential Eqs. (1)–(4) are transformed into a system of ordinary differential Eqs. (8)–(10) using the similarity transformations (7). The transformed Eqs. (8)–(10) subjected to the boundary conditions (11) were solved numerically using a Hermite wavelet technique with the help of MATHEMATICA software. The details of the method can be found [47,48,49, 51,52,53,54]. The results are given to carry out a parametric study showing the influences of the non-dimensional parameters, namely the cross diffusion terms, Hartmann number, thermophoresis parameter, squeeze number, Prandtl number and suction/injection parameter. The calculation of skin friction using the Hermite wavelet numerical technique has been performed for various values of $S$. The obtained results are compared with those obtained through other numerical methods by Fardi et al. [60]. The comparisons are presented in Tables 1,2,3,4,5,6, reveals a of consistency between the results obtained through the Hermite wavelet technique and high level those obtained through the other numerical techniques. Hence we can fulfil that HWT is a more suitable method for solving highly nonlinear coupled differential Eqs. (8)–(10) than any other techniques existing in the literature.

Table 1 Justification of results with Fardi et al. [60]

Table 2 $f^{\prime \prime}(1)$ validation using $\left( {S,\,\,M} \right) = \left( {0.1,\,\,0.2} \right)$

Table 3 $f^{\prime \prime}(1)$ validation using $\left( {A,\,\,M} \right) = \left( {0.1,\,\,0.2} \right)$

Table 4 $f^{\prime \prime}(1)$ validation using $\left( {A,\,\,S} \right) = \left( {0.1,\,\,0.1} \right)$

Table 5 The suction flow of $f^{\prime \prime}(1)$

Table 6 The injection flow of  $f^{\prime \prime}(1)$

The variations in the suction parameter $A>0$ and its impact on $f\left(\eta \right)$ and $f^{\prime}(\eta )$ can be observed in Figs. 1a, b and 2a, b respectively. For suction case the fluid flow is confined between two discs and emerges from the bottom disk. The output speed is greater than the bottom disk’s speed for a considerable amount of parameter $A$ without altering the top disk’s vertical flow. Increasing parameter $A$ leads to an increase in the suction power. Consequently, the direction of the axial flow changes (Fig. 2a, b). For smaller values of $A (0.1, 0.3),$ the values of $f(\eta )$ and $f{\prime}(\eta )$ are positive, whereas larger values of $A$ correspond to increased suction power, resulting in negative values for$f{\prime}(\eta )$. The conservation of mass is described by these negative values. The velocities on both the upper and lower disks are zero, as depicted in Fig. 2a, b. Figures 3a, b and 4a, b demonstrate that the axial velocity rises while the vertical velocity decreases as the parameter $S$ increases. The velocity of the top disc also increases with an increase in the parameter value$A$. Furthermore, as $S$ is increased, the highest axial velocity shifts from the centre to the bottom disk.

Fig. 1

Influence of $A$ on $f(\eta )$

Fig. 2

Influence of $A$ on $f^{\prime}(\eta )$

Fig. 3

Influence of $S$ on $f(\eta )$

Fig. 4

Influence of $S$ on $f^{\prime}(\eta )$

The strength of the magnetic field is studied by the Hartmann number, denoted as $M$, which significantly influences the behavior of fluid flow between two discs. In the case of parameter $A$, the axial flow direction transitions from the side to the centre, resulting in a negative axial velocity. Increasing $M$ reduces the axial velocity of the flow particles since it acts perpendicular to this velocity. However, the vertical velocity remains unaffected by the increase in $M$, as depicted in Figs. 5a, b and 6a, b. Figures 7a, b and 8a, b show how variational parameter A affects axial and vertical velocities, respectively. The fluid flow is introduced between the two discs and it is anticipated that the vertical velocity, as illustrated in Fig. 7a, b, will be zero somewhere between the two disks. The vertical velocity is zero close to the centre distance due to the higher entrance velocity from the bottom disc (bigger A). It can go farther because the input fluid flow has greater energy than the top disk. Thus, the fluid's vertical velocity achieves zero after colliding with the upstream flow and stabilizing the kinetic energy. From Fig. 8a, b, it is evident that a higher axial velocity (greater $A$) leads to an increase in the inlet flow’s flow rate. An increase in the axial output flow rate is required for mass conservation and incompressible fluid flow. Figures 9a, b and 10a, b provide visual representation of how the parameter $S$ influences the axial and vertical velocities, respectively. It can be observed that increasing $S$ leads to a decrease in vertical velocity over a specific distance. Figures 11a, b and 12a, b illustrates the vertical and axial flow velocities for different values of Hartmann number.

Fig. 5

Influence of $M$ on $f(\eta )$

Fig. 6

Influence of $M$ on $f^{\prime}(\eta )$

Fig. 7

Influence of $A$ on $f(\eta )$

Fig. 8

Influence of $A$ on $f^{\prime}(\eta )$

Fig. 9

Influence of $S$ on $f(\eta )$

Fig. 10

Influence of $S$ on $f^{\prime}(\eta )$

Fig. 11

Influence of $M$ on $f(\eta )$

Fig. 12

Influence of $M$ on $f^{\prime}(\eta )$

The influence of changes in the squeezing number on the temperature profile is depicted in Fig. 13. As the squeezing number increases the temperature decreases. Figure 14 demonstrates the decrease in temperature as the Prandtl number decreases. Furthermore, Fig. 15 shows that an increases in the Dufour quantity leads to an increases in the temperature profile. The impact of the squeezing number on the concentration distribution is illustrated in Fig. 16. Additionally, increasing Schmidt number values result in a decrease in concentration, as observed in Figs. 17, 18. These figures depict how the concentration curve increases with an increase in the Soret quantity.

Fig. 13

Influence of $S$ on $\theta (\eta )$

Fig. 14

Influence of $\Pr$ on $\theta (\eta )$

Fig. 15

Influence of $\gamma_{12}$ on $\theta (\eta )$

Fig. 16

Influence of $S$ on $\phi (\eta )$

Fig. 17

Influence of $Sc$ on $\phi (\eta )$

Fig. 18

Influence of ${\gamma }_{21}$ on $\phi (\eta )$

6 Conclusion

We have investigated the impact of cross diffusion on double diffusive MHD fluid flow using the Hermite wavelet method. The Hermite wavelet numerical solutions to the present problem were found to agree excellently with previously reported results for the single diffusive case. The influence of the governing parameters on the temperature, velocity, concentration and skin friction coefficient were examined and discussed. By analyzing the obtained results, we discovered several noteworthy findings with significant implications.

• The velocity near the upper plate region increases as the surfaces get closer jointly $(S<0)$, and it drops as they get additional apart $(S>0).$

• The velocity increases with increasing Hartmann number.

• The temperature and concentrations are decreases with increasing Hartmann number.

• An elevation of both Hartmann parameter $(M)$ and Prandtl number $(Pr)$ leads to an increase in temperature and the rate of heat transfer.

• The temperature decreases with increasing squeezing number.

• Increase in the Dufour quantity leads to an increase the temperature profile.

• The concentration profile increases with an increase in the Soret quantity.