Thermohaline convection in MHD Casson fluid over an exponentially stretching sheet

The flow over a stretching surface is an important problem in many engineering processes with applications in industries such as extrusion, glass-fiber production, hot rolling, melt-spinning, wire drawing, manufacture of plastic and rubber sheets, cooling of a large metallic plate in a bath, which may be an electrolyte, etc [1, 2]. Experiments show that the velocity of the stretching surface is approximately proportional to the distance from the orifice [3]. Crane [4] studied the steady two-dimensional incompressible boundary layer flow of a Newtonian fluid caused by the stretching sheet. Nazar et al. [5] studied boundary layer flow in the region of the stagnation point on a stretching sheet. Fang et al. [6] examined boundary layer flow over a stretching sheet with variable thickness.

In recent years, some interest has been given to the study of convective transport of Non-Newtonian fluids. Non-Newtonian fluids encompass a captivating and diverse group of fluids that present behaviours distinct from the conventional Newtonian fluid model. Diverging from Newtonian fluids, which maintain a consistent viscosity irrespective of the applied shear stress, non-Newtonian fluids possess viscosities that fluctuate in response to external forces. The study of non-Newtonian fluids also has applications in biomedical engineering, geophysics, rheology, and many other fields [7,8,9,10]. Casson fluid is one such type of non-Newtonian fluid that is widely used for modelling blood flow in narrow arteries. Bhattacharyya [11] studied the research which examines the stable boundary layer stagnation-point flow and heat transmission of Casson fluid on a sheet undergoing contraction or expansion. Mustafa et al. [12] investigate the time-varying behavior of the flow and heat transfer of a non-Newtonian fluid (Casson fluid) over a moving flat plate that is exposed to a parallel free stream. Stagnation-point flow and heat transfer of a Casson fluid towards a stretching sheet is studied by Mustafa et al. [13]. Mukhopadhyay et al. [14] examined the flow and heat transmission of a Casson past a symmetric wedge, under the forced convection regime.

However, thermohaline convection is a phenomenon that occurs when there are gradients in both temperature and concentration in a fluid. It is a significant topic in fluid mechanics and has several real-life applications [15,16,17]. Verma and Mondal [18] reviewed heat and mass transfer of Casson fluids. Ibrahim et al. [19] investigated how the flow of a non-Newtonian (Casson fluid) is affected by a heat source, suction/injection, and chemical reaction when it passes over an exponentially stretching sheet. Aghighi et al. [20] examined double-diffusive natural convection of Casson fluids in an enclosure.

Magnetohydrodynamics (MHD) is extensively used as a tool for studying fluid dynamics in general. MHD plays a crucial role in understanding natural phenomena in space propulsion [21], industrial applications [22] and geophysics and astrophysics [23]. It helps in understanding the behavior of conducting fluids in the presence of magnetic fields, including the interaction of magnetic fields with fluid flow [24,25,26]. Moreover, a non-uniform heat source/sink refers to a situation where the heat generation or absorption within a system is not evenly distributed. This phenomenon has significant implications in various fields, including thermal engineering [27], industrial processes [28] and environmental engineering [29]. The combination of double diffusive convection, non-uniform heat source and magnetic field has a wide range of applications [30,31,32]. A study by Nadeem et al. [33] examined the MHD boundary layer flow of a fluid with Casson rheology over a shrinking sheet that has an exponential permeability. Mukhopadhyay et al. [34] used a computer simulation to study the flow of non-Newtonian (Casson fluid) over a sheet that is stretching exponentially, in the existence of a magnetic flux. Ghilasi and Saleh [35] studied the Casson flow over a stationary plate in the influence of a magnetic flux and a non-uniform heat source/sink using the homotopy analysis method. Bhatti et al. [36] obtained differential transform solution for Hall and ion-slip effects on radiative-convective Casson flow from a stretching sheet with convective heating. Ganesh and Sridhar [37] examined the double convection characteristics of a MHD non-Newtonian fluid over an exponentially porous stretching sheet.

Mathematical modeling of various phenomena in diverse fields such as engineering, science and technology often involves nonlinear differential equations. Such nonlinear equations naturally arise in many real-world problems pertaining to fluid dynamics, materials science, physics, biology and other domains [38,39,40]. However, analytical solutions are rarely feasible for most nonlinear problems except trivial cases. As a result, numerical methods are commonly employed to obtain approximate solutions. In recent years, several numerical techniques have been proposed for solving highly nonlinear differential equations, including the Runge-Kutta method, finite difference approaches, optimal homotopy analysis, homotopy perturbation method, and numerical wavelet approaches. Numerical wavelet methods have gained significant attention in recent years for solving differential equations arising in diverse fields such as signal processing, image analysis and mathematical modeling. Notable contributions include the use of Laguerre wavelets [41], Taylor wavelets [42], cardinal B-splines [43], Bernoulli wavelets [44] and Hermite wavelets [45, 46]. Among these, the HWM has been widely utilized due to its accuracy. Given the effectiveness of numerical wavelet techniques, developing a parametric Hermite wavelet approach for nonlinear problems is important. With the availability of powerful computational software, Hermite wavelet methods have been successfully employed by researchers to solve various nonlinear boundary value problems, including different fluid flow models [47,48,49,50,51,52,53].

In this paper, we apply the HWM to analyze MHD double diffusive non-Newtonian (Casson fluid) fluid flow over a sheet with a variable heat source or sink. Our numerical wavelet results are more effective and accurate than other numerical techniques. The impact of influential physical parameters such as the Prandtl number, Schmidt number, Magnetic field, suction or blowing, chemical reaction and radiation parameters on the velocity, temperature and concentration distributions in the boundary layer are examined.

The paper is organized as follows: Section 2 introduces the mathematical model of the problem. Sections 3 and 4 present the methodology of HWM and the integration of matrix, respectively. Section 5 presents the method of solution procedure. Section 6 discusses the results. Section 7 provides the conclusions of this paper.

We examine a steady, laminar boundary layer flow of an incompressible Casson fluid over an exponentially expanding surface aligned with the y = 0 plane, with the fluid flow constrained to y > 0. This flow increases from the linear stretching of the surface, induced by the concurrent application of two equal and opposite forces along the x-axis. Preserving the original configuration, the sheet stretches at velocity U_w(x). The lower and upper boundaries are held at constant temperatures T(w) and T_∞, respectively, while solute concentration is held at fixed values C(w) and C_∞, respectively, as represented in Figure 1. Within the common boundary layer flow approximations, heat and mass transfer characteristics of the Casson fluid can be described by the following equations for continuity, momentum, energy and concentration in Cartesian coordinates x and y [54, 55] (1) ∂u∂x+∂u∂y=0, {{\partial u} \over {\partial x}} + {{\partial u} \over {\partial y}} = 0, (2) u∂u∂x+v∂u∂y=ν(1+1β)∂2u∂y2−σB2uρ, u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = \nu \left( {1 + {1 \over \beta}} \right){{{\partial^2}u} \over {\partial {y^2}}} - {{\sigma {B^2}u} \over \rho}, (3) u∂T∂x+v∂T∂y=κTρCp∂2T∂y2−1ρCp∂qr∂y+QρCp(T−T∞), u{{\partial T} \over {\partial x}} + v{{\partial T} \over {\partial y}} = {{{\kappa_T}} \over {\rho {C_p}}}{{{\partial^2}T} \over {\partial {y^2}}} - {1 \over {\rho {C_p}}}{{\partial {q_r}} \over {\partial y}} + {Q \over {\rho {C_p}}}\left( {T - {T_\infty}} \right), (4) u∂C∂x+v∂C∂y=κCρCp∂2C∂y2−K1(C−C∞), u{{\partial C} \over {\partial x}} + v{{\partial C} \over {\partial y}} = {{{\kappa_C}} \over {\rho {C_p}}}{{{\partial^2}C} \over {\partial {y^2}}} - {K_1}\left( {C - {C_\infty}} \right), where x-axis velocity (u), y-axis velocity (v), density of fluid (ρ), electrical conductivity (σ), kinematic viscosity(ν), specific heat at constant pressure (C_p), temperature (T), thermal diffusivity (κ_T), dimensional heat source(Q), solute diffusivity (κ_C), chemical reaction(K₁), radiative heat flux (q_r), Casson parameter (β) and magnetic field(B).

Fig. 1

Physical configuration.

Using the Rossel and approximation for radiation, we get (5) qr=−4σ3K*∂T4∂y,T4≈4T∞3T−3T∞4 ,qr=−4σ3K*∂(4T∞3T−3T∞4)∂y=−16σT∞33K*∂T∂y. \matrix{{{q_r} = - {{4\sigma} \over {3{K^*}}}{{\partial {T^4}} \over {\partial y}},{T^4} \approx 4T_\infty^3T - 3T_\infty^4\;,} \cr {{q_r} = - {{4\sigma} \over {3{K^*}}}{{\partial \left( {4T_\infty^3T - 3T_\infty^4} \right)} \over {\partial y}} = - {{16\sigma T_\infty^3} \over {3{K^*}}}{{\partial T} \over {\partial y}}.} \cr}

Hence, (3) reduce to (6) u∂T∂x+v∂T∂y=κTρCp∂2T∂y2+16σT∞33K*1ρCp∂2T∂y2+QρCp(T−T∞). u{{\partial T} \over {\partial x}} + v{{\partial T} \over {\partial y}} = {{{\kappa_T}} \over {\rho {C_p}}}{{{\partial^2}T} \over {\partial {y^2}}} + {{16\sigma T_\infty^3} \over {3{K^*}}}{1 \over {\rho {C_p}}}{{{\partial^2}T} \over {\partial {y^2}}} + {Q \over {\rho {C_p}}}\left( {T - {T_\infty}} \right).

The boundary conditions are (7) u=Uw(x), v=−V(x), T=Tw, C=Cw at y=0,u=0, T=T∞, C=C∞ at y→∞. \matrix{{u = {U_w}\left( x \right),\;v = - V\left( x \right),\;T = {T_w},{\kern 1pt} C = {C_w}{\kern 1pt} {\kern 1pt} {\kern 1pt} at\;{\kern 1pt} {\kern 1pt} {\kern 1pt} y = 0,} \hfill \cr {\;u = 0,\;T = {T_\infty},{\kern 1pt} C = {C_\infty}\;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} at{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;y \to \infty.} \hfill \cr}

The wall and the boundary layer's outer edge have different conditions, which are denoted by the subscripts w and ∞.

Introducing the following similarity variables as (8) η=yex2Lu02νL, B=B0ex2L, u=u0exLf′(η), v=−ex2Lu02νL(f(η)+ηf′(η))T=T∞+T0ex2Lθ(η), C=C∞+C0ex2Lφ(η), Q=Q0ex2L, K1=k0ex2L. \matrix{{\eta = y{e^{{x \over {2L}}}}\sqrt {{{{u_0}} \over {2\nu L}}},{\kern 1pt} {\kern 1pt} B = {B_0}{e^{{x \over {2L}}}},\;\;u = {u_0}{e^{{x \over L}}}f'\left( \eta \right),{\kern 1pt} {\kern 1pt} v = - {e^{{x \over {2L}}}}\sqrt {{{{u_0}} \over {2\nu L}}} \left( {f(\eta ) + \eta f'\left( \eta \right)} \right)} \hfill \cr {T = {T_\infty} + {T_0}{e^{{x \over {2L}}}}\theta \left( \eta \right),\;{\kern 1pt} {\kern 1pt} C = {C_\infty} + {C_0}{e^{{x \over {2L}}}}\varphi \left( \eta \right),\;{\kern 1pt} {\kern 1pt} Q = {Q_0}{e^{{x \over {2L}}}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {K_1} = {k_0}{e^{{x \over {2L}}}}} \hfill \cr}.

Substitute (8) in (1) to (6) we have, (9) (1+1β)f′′′+ff′′−2f′2−Mf′=0, \left( {1 + {1 \over \beta}} \right)f''' + ff'' - 2{f'^2} - Mf' = 0, (10) (1+43R)θ′′+Pr(fθ′−f′θ)+PrHsθ=0, \left( {1 + {4 \over 3}R} \right)\theta '' + \Pr \left( {f\theta ' - f'\theta} \right) + \Pr Hs\theta = 0, (11) φ′′+Sc(fφ′−f′φ)−2γφ=0. \varphi '' + Sc\left( {f\varphi ' - f'\varphi} \right) - 2\gamma \varphi = 0.

The corresponding boundary conditions become, (12) f(0)=S, f′(0)=1, f′(∞)→0;θ(0)=1,θ(∞)→0 ;φ(0)=1,φ(∞)→0 . f\left( 0 \right) = S,\;f'\left( 0 \right) = 1,\;f'\left( \infty \right) \to 0;\theta \left( 0 \right) = 1,\theta \left( \infty \right) \to 0\;;\varphi \left( 0 \right) = 1,\varphi \left( \infty \right) \to 0.

Where, Pr=μCpκT \Pr = {{\mu {C_p}} \over {{\kappa_T}}} is the Prandtl number, M=σB022Lu0ρ M = {{\sigma B_0^22L} \over {{u_0}\rho}} is the magnetic field parameter, S=v0u0ν2L S = {{{v_0}} \over {\sqrt {{{{u_0}\nu} \over {2L}}}}} is the suction or blowing parameter, Sc=μCpκC Sc = {{\mu {C_p}} \over {{\kappa_C}}} is the Schmidt number, γ=k0Lu0 \gamma = {{{k_0}L} \over {{u_0}}} is the chemical reaction parameter, R= 4σT∞3κTK* R = {{4\sigma T_\infty^3} \over {{\kappa_T}{K^*}}} is the radiation parameter, Hs=Q0ρCp2Lu0ρ Hs = {{{Q_0}} \over {\rho {C_p}}}{{2L} \over {{u_0}\rho}} is the source/sink parameter. Via Table 1, the physical meanings of parameters are given.

Hermite wavelets are well-defined as [45, 47], ϕn.m(x)={2k−12πHm(2kx−2n+1) , n−12k−1≤x≤n2k−10 elsewhere. {\phi_{n.m}}(x) = \left\{{\matrix{{{{{2^{{{^{k - 1}} \over 2}}}} \over {\sqrt \pi}}{H_m}\left( {{2^k}x - 2n + 1} \right){\kern 1pt},{\kern 1pt} {\kern 1pt} {\kern 1pt} {{n - 1} \over {{2^{k - 1}}}} \le x \le {n \over {{2^{k - 1}}}}} \hfill \cr {0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm elsewhere}} \hfill \cr}} \right..

Where n = 1, 2, 3, ⋯, 2^k−1 and m = 0, 1, 2, 3, ⋯, M − 1 and k ∈ N Here H_m(x) represents Hermite polynomials of degree m with regards to weight function W(x)=1−x2 W(x) = \sqrt {1 - {x^2}} on the real line R and pleases the succeeding recurrence formula H₀(x) = 1, H₁(x) = 2x, H_m+2(x) = 2xH_m+1(x) − 2(m + 1) H_m(x) where, m = 0, 1, 2, 3,⋯.

For a specified nonlinear differential equation, we approximated to the solution as y(x)with the assistance of HWM as follows y(x)=∑n=1∞∑m=0∞Cn,mϕn,m(x). y(x) = \sum\limits_{n = 1}^\infty \sum\limits_{m = 0}^\infty {C_{n,m}}{\phi_{n,m}}(x).

We estimated y(x) by truncating the series as follows y(x)≈∑n=12k−1∑m=0M−1Cn,mϕn,m(x)=ATϕ(x), y(x) \approx \sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {C_{n,m}}{\phi_{n,m}}(x) = {A^T}\phi (x), where A and ϕ(x) are 2^k−1M × 1 matrix, AT=[C1,0,⋯C1,M−1,C2,0⋯C2,M−1,⋯C2k−1,0,⋯C2k−1,M−1],.ϕ(x)=[ϕ1,0,⋯ϕ1,M−1,ϕ2,0,⋯ϕ2,M−1,⋯ϕ2k−1,0,⋯ϕ2k−1,M−1]T. \matrix{{{A^T} = [{C_{1,0}}, \cdots {C_{1,M - 1}},{C_{2,0}} \cdots {C_{2,M - 1}}, \cdots {C_{{2^{k - 1}},0}}, \cdots {C_{{2^{k - 1}},M - 1}}],.} \cr {\phi (x) = {{[{\phi_{1,0}}, \cdots {\phi_{1,M - 1}},{\phi_{2,0}}, \cdots {\phi_{2,M - 1}}, \cdots {\phi_{{2^{k - 1}},0}}, \cdots {\phi_{{2^{k - 1}},M - 1}}]}^T}.} \cr}

Let {ϕ_{i, j}} be the system of Hermite wavelets, n=1, 2, ⋯ and m = 0, 1, ⋯. For each fixed n, there is a Hermite space generated by the elements of the sequence {ϕ_{i, j}}. That is, L ({ϕ_{i, j}}) = H²[0, 1) is Banach space. This article is associated to some essential theorems that can be seen in [45, 47].

The following are some of the Hermite wavelet basis at k = 1 ϕ1,0(x)=2π,ϕ1,1(x)=1π(8x−4),ϕ1,2(x)=1π(32x2−32x+4),ϕ1,3(x)=1π(128x3−192x2+48x+8),ϕ1,4(x)=1π(512x4−1024x3+384x2+128x−40), \matrix{{{\phi_{1,0}}(x) = {2 \over {\sqrt \pi}},} \hfill \cr {{\phi_{1,1}}(x) = {1 \over {\sqrt \pi}}(8x - 4),} \hfill \cr {{\phi_{1,2}}(x) = {1 \over {\sqrt \pi}}(32{x^2} - 32x + 4),} \hfill \cr {{\phi_{1,3}}(x) = {1 \over {\sqrt \pi}}(128{x^3} - 192{x^2} + 48x + 8),} \hfill \cr {{\phi_{1,4}}(x) = {1 \over {\sqrt \pi}}(512{x^4} - 1024{x^3} + 384{x^2} + 128x - 40),} \hfill \cr} ϕ1,5(x)=1π(2048x5−5120x4+2560x3+1280x2−800x+16),ϕ1,6(x)=1π(8192x6−24576x5+15360x4+10240x3−9600x2+384x+368),ϕ1,7(x)=1π(32768x7−114688x6+86016x5+71680x4−89600x3+5376x2+10304x−928),ϕ1,8(x)=1π(131072x8−524288x7+458752x6+458752x5−716800x4+ 57344x3+164864x2−29696x−3296),ϕ1,9(x)=1π(524288x9−2359296x8+2359296x7+2752512x6−5160960x5+516096x4+ 1978368x3−534528x2−118656x+21440),ϕ1,10(x)=1π(2097152x10−10485760x9+11796480x8+15728640x7−34406400x6+ 4128768x5+19783680x4−7127040x3−2373120x2+857600x+16448),ϕ1,11(x)=1π(8388608x11−46137344x10+57671680x9+86507520x8−216268800x7+ 30277632x6+174096384x5−78397440x4−34805760x3+18867200x2 , +723712x−461696)ϕ1,12(x)=1π(33554432x12−201326592x11+276824064x10+461373440x9−1297612800x8 +207618048x7+1392771072x6−752615424x5−417669120x4+301875200x3 +17369088x2−22161408x+561536), \matrix{{{\phi_{1,5}}(x) = {1 \over {\sqrt \pi}}(2048{x^5} - 5120{x^4} + 2560{x^3} + 1280{x^2} - 800x + 15),} \cr {{\phi_{1,6}}(x) = {1 \over {\sqrt \pi}}(8192{x^6} - 24576{x^5} + 15360{x^4} + 10240{x^3} - 9600{x^2} + 384x + 368),} \cr {{\phi_{1,7}}(x) = {1 \over {\sqrt \pi}}(32768{x^7} - 114688{x^6} + 86016{x^5} + 71680{x^4} - 89600{x^3} + 5376{x^2} + 10304x - 928),} \cr {\matrix{{{\phi_{1,8}}(x) = {1 \over {\sqrt \pi}}(131072{x^8} - 524288{x^7} + 458752{x^6} + 458752{x^5} - 716800{x^4} +} \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 57344{x^3} + 164864{x^2} - 29696x - 3296)} \hfill \cr},} \cr {\matrix{{{\phi_{1,9}}(x) = {1 \over {\sqrt \pi}}(524288{x^9} - 2359296{x^8} + 2359296{x^7} + 2752512{x^6} - 5160960{x^5} + 516096{x^4} +} \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1978368{x^3} - 534528{x^2} - 118656x + 21440)} \hfill \cr},} \cr {\matrix{{{\phi_{1,10}}(x) = {1 \over {\sqrt \pi}}(2097152{x^{10}} - 10485760{x^9} + 11796480{x^8} + 15728640{x^7} - 34406400{x^6} +} \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 4128768{x^5} + 19783680{x^4} - 7127040{x^3} - 2373120{x^2} + 857600x + 16448)} \hfill \cr},} \cr {\matrix{{{\phi_{1,11}}(x) = {1 \over {\sqrt \pi}}(8388608{x^{11}} - 46137344{x^{10}} + 57671680{x^9} + 86507520{x^8} - 216268800{x^7} +} \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 30277632{x^6} + 174096384{x^5} - 78397440{x^4} - 34805760{x^3} + 18867200{x^2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt},} \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + 723712x - 461696)} \hfill \cr}} \cr {\matrix{{{\phi_{1,12}}(x) = {1 \over {\sqrt \pi}}(33554432{x^{12}} - 201326592{x^{11}} + 276824064{x^{10}} + 461373440{x^9} - 1297612800{x^8}} \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + 207618048{x^7} + 1392771072{x^6} - 752615424{x^5} - 417669120{x^4} + 301875200{x^3}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}} \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + 17369088{x^2} - 22161408x + 561536),} \hfill \cr}} \cr} where, ϕ₉(x) = [ϕ_1,0(x), ϕ_1,1(x), ϕ_1,2(x), ϕ_1,3(x), ϕ_1,4(x), ϕ_1,5(x), ϕ_1,6(x), ϕ_1,7(x), ϕ_1,8(x)]^T.

Apply the integration regarding with x and limits from 0 to x the above nine bases and explicit it as a linear combination of Hermite wavelet basis as follows. ∫0xϕ1,0(x)=[12140000000] ϕ9( x),∫0xϕ1,1( x)=[−14018000000] ϕ9(x),∫0xϕ1,2(x)=[−130011200000] ϕ9(x),∫0xϕ1,3(x)=[540001160000] ϕ9(x),∫0xϕ1,4(x)=[−250000120000] ϕ9(x),∫0xϕ1,5(x)=[−2330000012400] ϕ9(x),∫0xϕ1,6(x)=[11670000001280] ϕ9(x),∫0xϕ1,7(x)=[10320000000132] ϕ9(x),∫0xϕ1,8(x)=[−2680900000000] ϕ9(x)+136ϕ1,9 (x). \matrix{{\int_0^x {\phi_{1,0}}(x) = \left[{\matrix{{{1 \over 2}} & {{1 \over 4}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\kern 1pt} {\phi_9}({\kern 1pt} x),} \cr {\int_0^x {\phi_{1,1}}({\kern 1pt} x) = \left[{\matrix{{{{- 1} \over 4}} & 0 & {{1 \over 8}} & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x {\phi_{1,2}}(x) = \left[{\matrix{{{{- 1} \over 3}} & 0 & 0 & {{1 \over {12}}} & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x {\phi_{1,3}}(x) = \left[{\matrix{{{5 \over 4}} & 0 & 0 & 0 & {{1 \over {16}}} & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x {\phi_{1,4}}(x) = \left[{\matrix{{{{- 2} \over 5}} & 0 & 0 & 0 & 0 & {{1 \over {20}}} & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x {\phi_{1,5}}(x) = \left[{\matrix{{{{- 23} \over 3}} & 0 & 0 & 0 & 0 & 0 & {{1 \over {24}}} & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x {\phi_{1,6}}(x) = \left[{\matrix{{{{116} \over 7}} & 0 & 0 & 0 & 0 & 0 & 0 & {{1 \over {28}}} & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x {\phi_{1,7}}(x) = \left[{\matrix{{{{103} \over 2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{1 \over {32}}} \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x {\phi_{1,8}}(x) = \left[{\matrix{{{{- 2680} \over 9}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x) + {1 \over {36}}{\phi_{1,9}}{\kern 1pt} (x).} \cr}

Hence, ∫0x ϕ( x) dx =H9×9 ϕ9(x)+ ϕ¯9 (x), \int_0^x {\kern 1pt} \phi ({\kern 1pt} x){\kern 1pt} dx{\kern 1pt} = {H_{9 \times 9}}{\kern 1pt} {\kern 1pt} {\phi_9}(x) + {\kern 1pt} {\kern 1pt} {\bar \phi_9}{\kern 1pt} (x), where H9×9=[12140000000−14018000000−130011200000540001160000−250000120000−23300000124001167000000128010320000000132−2680900000000], ϕ9¯(x)=[00000000136ϕ1,9(x)]. {H_{9 \times 9}} = \left[{\matrix{{{1 \over 2}} & {{1 \over 4}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr {{{- 1} \over 4}} & 0 & {{1 \over 8}} & 0 & 0 & 0 & 0 & 0 & 0 \cr {{{- 1} \over 3}} & 0 & 0 & {{1 \over {12}}} & 0 & 0 & 0 & 0 & 0 \cr {{5 \over 4}} & 0 & 0 & 0 & {{1 \over {16}}} & 0 & 0 & 0 & 0 \cr {{{- 2} \over 5}} & 0 & 0 & 0 & 0 & {{1 \over {20}}} & 0 & 0 & 0 \cr {{{- 23} \over 3}} & 0 & 0 & 0 & 0 & 0 & {{1 \over {24}}} & 0 & 0 \cr {{{116} \over 7}} & 0 & 0 & 0 & 0 & 0 & 0 & {{1 \over {28}}} & 0 \cr {{{103} \over 2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{1 \over {32}}} \cr {{{- 2680} \over 9}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right],{\kern 1pt} {\kern 1pt} \overline {{\phi_9}} (x) = \left[{\matrix{0 \cr 0 \cr 0 \cr 0 \cr 0 \cr 0 \cr 0 \cr 0 \cr {{1 \over {36}}{\phi_{1,9}}(x)} \cr}} \right].

Implement the integration to the above nine bases is as follows ∫0x∫0xϕ1,0(x) dxdx=[31618132000000] ϕ9(x),∫0x∫0xϕ1,1(x) dxdx=[−16−116019600000] ϕ9(x),∫0x∫0xϕ1,2(x) dxdx=[−116−1120011920000] ϕ9(x),∫0x∫0xϕ1,3(x) dxdx=[355160001320000] ϕ9(x),∫0x∫0xϕ1,4(x) dxdx=[−712−1100000148000] ϕ9(x),∫0x∫0xϕ1,5(x) dxdx=[−227−23120000016720] ϕ9(x),∫0x∫0xϕ1,6(x) dxdx=[8182970000001896] ϕ9(x),∫0x∫0xϕ1,7(x) dxdx=[148910380000000] ϕ9(x)+11152ϕ1,9(x),∫0x∫0xϕ1,8(x) dxdx=[−7735−67090000000] ϕ9(x)+11440ϕ1,10(x). \matrix{{\int_0^x \int_0^x {\phi_{1,0}}(x){\kern 1pt} dxdx = \left[{\matrix{{{3 \over {16}}} & {{1 \over 8}} & {{1 \over {32}}} & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x {\phi_{1,1}}(x){\kern 1pt} dxdx = \left[{\matrix{{{{- 1} \over 6}} & {{{- 1} \over {16}}} & 0 & {{1 \over {96}}} & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x {\phi_{1,2}}(x){\kern 1pt} dxdx = \left[{\matrix{{{{- 1} \over {16}}} & {{{- 1} \over {12}}} & 0 & 0 & {{1 \over {192}}} & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x {\phi_{1,3}}(x){\kern 1pt} dxdx = \left[{\matrix{{{3 \over 5}} & {{5 \over {16}}} & 0 & 0 & 0 & {{1 \over {320}}} & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x {\phi_{1,4}}(x){\kern 1pt} dxdx = \left[{\matrix{{{{- 7} \over {12}}} & {{{- 1} \over {10}}} & 0 & 0 & 0 & 0 & {{1 \over {480}}} & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x {\phi_{1,5}}(x){\kern 1pt} dxdx = \left[{\matrix{{{{- 22} \over 7}} & {{{- 23} \over {12}}} & 0 & 0 & 0 & 0 & 0 & {{1 \over {672}}} & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x {\phi_{1,6}}(x){\kern 1pt} dxdx = \left[{\matrix{{{{81} \over 8}} & {{{29} \over 7}} & 0 & 0 & 0 & 0 & 0 & 0 & {{1 \over {896}}} \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x {\phi_{1,7}}(x){\kern 1pt} dxdx = \left[{\matrix{{{{148} \over 9}} & {{{103} \over 8}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x) + {1 \over {1152}}{\phi_{1,9}}(x),} \cr {\int_0^x \int_0^x {\phi_{1,8}}(x){\kern 1pt} dxdx = \left[{\matrix{{{{- 773} \over 5}} & {{{- 670} \over 9}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x) + {1 \over {1440}}{\phi_{1,10}}(x).} \cr}

Hence, ∫0x∫0x ϕ(x) dx dx= H′9×9 ϕ9(x)+ ϕ¯′9(x), \int_0^x \int_0^x {\kern 1pt} \phi (x){\kern 1pt} {\kern 1pt} dx{\kern 1pt} dx = {\kern 1pt} {H'_{9 \times 9}}{\kern 1pt} {\kern 1pt} {\phi_9}(x) + {\kern 1pt} {\kern 1pt} {\kern 1pt} \bar \phi {'_9}(x), where H′9×9=[31618132000000−16−116019600000−116−1120011920000355160001320000−712−1100000148000−227−231200000167208182970000001896148910380000000−7735−67090000000], ϕ′9¯(x)=[000000011152ϕ1,9(x)11440ϕ1,10 (x)]. {H'_{9 \times 9}} = \left[{\matrix{{{3 \over {16}}} & {{1 \over 8}} & {{1 \over {32}}} & 0 & 0 & 0 & 0 & 0 & 0 \cr {{{- 1} \over 6}} & {{{- 1} \over {16}}} & 0 & {{1 \over {96}}} & 0 & 0 & 0 & 0 & 0 \cr {{{- 1} \over {16}}} & {{{- 1} \over {12}}} & 0 & 0 & {{1 \over {192}}} & 0 & 0 & 0 & 0 \cr {{3 \over 5}} & {{5 \over {16}}} & 0 & 0 & 0 & {{1 \over {320}}} & 0 & 0 & 0 \cr {{{- 7} \over {12}}} & {{{- 1} \over {10}}} & 0 & 0 & 0 & 0 & {{1 \over {480}}} & 0 & 0 \cr {{{- 22} \over 7}} & {{{- 23} \over {12}}} & 0 & 0 & 0 & 0 & 0 & {{1 \over {672}}} & 0 \cr {{{81} \over 8}} & {{{29} \over 7}} & 0 & 0 & 0 & 0 & 0 & 0 & {{1 \over {896}}} \cr {{{148} \over 9}} & {{{103} \over 8}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr {{{- 773} \over 5}} & {{{- 670} \over 9}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right],{\kern 1pt} {\kern 1pt} \overline {{{\phi '}_9}} (x) = \left[{\matrix{0 \cr 0 \cr 0 \cr 0 \cr 0 \cr 0 \cr 0 \cr {{1 \over {1152}}{\phi_{1,9}}(x)} \cr {{1 \over {1440}}{\phi_{1,10}}{\kern 1pt} (x)} \cr}} \right].

Again, we employ the integration to the above nine bases as follows ∫0x∫0x∫0xϕ1,0(x) dxdxdx=[596364164138400000] ϕ9(x),∫0x∫0x∫0xϕ1,1(x) dxdxdx=[−7128−124−11280115360000] ϕ9(x),∫0x∫0x∫0xϕ1,2(x) dxdxdx=[−180−164−1960013840000] ϕ9(x),∫0x∫0x∫0xϕ1,3(x) dxdxdx=[199632051280001768000] ϕ9(x),∫0x∫0x∫0xϕ1,4(x) dxdxdx=[−1356−748−18000001134400] ϕ9(x),∫0x∫0x∫0xϕ1,5(x) dxdxdx=[−6564−1114−239600000121504] ϕ9(x),∫0x∫0x∫0xϕ1,6(x) dxdxdx=[1333681322956000000] ϕ9(x)+132256ϕ1,9(x),∫0x∫0x∫0xϕ1,7(x) dxdxdx=[1934037910364000000] ϕ9(x)+146080ϕ1,10(x),∫0x∫0x∫0xϕ1,8(x) dxdxdx=[−121122−77320−33536000000] ϕ9(x)+163360ϕ1,11(x). \matrix{{\int_0^x \int_0^x \int_0^x {\phi_{1,0}}(x){\kern 1pt} dxdxdx = \left[{\matrix{{{5 \over {96}}} & {{3 \over {64}}} & {{1 \over {64}}} & {{1 \over {384}}} & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x \int_0^x {\phi_{1,1}}(x){\kern 1pt} dxdxdx = \left[{\matrix{{{{- 7} \over {128}}} & {{{- 1} \over {24}}} & {{{- 1} \over {128}}} & 0 & {{1 \over {1536}}} & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x \int_0^x {\phi_{1,2}}(x){\kern 1pt} dxdxdx = \left[{\matrix{{{{- 1} \over {80}}} & {{{- 1} \over {64}}} & {{{- 1} \over {96}}} & 0 & 0 & {{1 \over {3840}}} & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x \int_0^x {\phi_{1,3}}(x){\kern 1pt} dxdxdx = \left[{\matrix{{{{19} \over {96}}} & {{3 \over {20}}} & {{5 \over {128}}} & 0 & 0 & 0 & {{1 \over {7680}}} & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x \int_0^x {\phi_{1,4}}(x){\kern 1pt} dxdxdx = \left[{\matrix{{{{- 13} \over {56}}} & {{{- 7} \over {48}}} & {{{- 1} \over {80}}} & 0 & 0 & 0 & 0 & {{1 \over {13440}}} & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x \int_0^x {\phi_{1,5}}(x){\kern 1pt} dxdxdx = \left[{\matrix{{{{- 65} \over {64}}} & {{{- 11} \over {14}}} & {{{- 23} \over {96}}} & 0 & 0 & 0 & 0 & 0 & {{1 \over {21504}}} \cr}} \right]{\kern 1pt} {\phi_9}(x),} \cr {\int_0^x \int_0^x \int_0^x {\phi_{1,6}}(x){\kern 1pt} dxdxdx = \left[{\matrix{{{{133} \over {36}}} & {{{81} \over {32}}} & {{{29} \over {56}}} & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x) + {1 \over {32256}}{\phi_{1,9}}(x),} \cr {\int_0^x \int_0^x \int_0^x {\phi_{1,7}}(x){\kern 1pt} dxdxdx = \left[{\matrix{{{{193} \over {40}}} & {{{37} \over 9}} & {{{103} \over {64}}} & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x) + {1 \over {46080}}{\phi_{1,10}}(x),} \cr {\int_0^x \int_0^x \int_0^x {\phi_{1,8}}(x){\kern 1pt} dxdxdx = \left[{\matrix{{{{- 1211} \over {22}}} & {{{- 773} \over {20}}} & {{{- 335} \over {36}}} & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right]{\kern 1pt} {\phi_9}(x) + {1 \over {63360}}{\phi_{1,11}}(x).} \cr}

Hence, ∫0x∫0x∫0x ϕ(x) dxdxdx=H′′9×9 ϕ9(x)+ ϕ¯″9(x), \int_0^x \int_0^x \int_0^x {\kern 1pt} \phi (x){\kern 1pt} dxdxdx = {H''_{9 \times 9}}{\kern 1pt} {\kern 1pt} {\phi_9}(x) + {\kern 1pt} {\kern 1pt} {\bar \phi ''_9}(x), where H′′9×9=[596364164138400000−7128−124−11280115360000−180−164−1960013840000199632051280001768000−1356−748−18000001134400−6564−1114−23960000012150413336813229560000001934037910364000000−121122−77320−33536000000],ϕ′′9¯(x)=[000000132256ϕ1,9(x)146080ϕ1,10(x)163360ϕ1,11(x)]. {H''_{9 \times 9}} = \left[{\matrix{{{5 \over {96}}} & {{3 \over {64}}} & {{1 \over {64}}} & {{1 \over {384}}} & 0 & 0 & 0 & 0 & 0 \cr {{{- 7} \over {128}}} & {{{- 1} \over {24}}} & {{{- 1} \over {128}}} & 0 & {{1 \over {1536}}} & 0 & 0 & 0 & 0 \cr {{{- 1} \over {80}}} & {{{- 1} \over {64}}} & {{{- 1} \over {96}}} & 0 & 0 & {{1 \over {3840}}} & 0 & 0 & 0 \cr {{{19} \over {96}}} & {{3 \over {20}}} & {{5 \over {128}}} & 0 & 0 & 0 & {{1 \over {7680}}} & 0 & 0 \cr {{{- 13} \over {56}}} & {{{- 7} \over {48}}} & {{{- 1} \over {80}}} & 0 & 0 & 0 & 0 & {{1 \over {13440}}} & 0 \cr {{{- 65} \over {64}}} & {{{- 11} \over {14}}} & {{{- 23} \over {96}}} & 0 & 0 & 0 & 0 & 0 & {{1 \over {21504}}} \cr {{{133} \over {36}}} & {{{81} \over {32}}} & {{{29} \over {56}}} & 0 & 0 & 0 & 0 & 0 & 0 \cr {{{193} \over {40}}} & {{{37} \over 9}} & {{{103} \over {64}}} & 0 & 0 & 0 & 0 & 0 & 0 \cr {{{- 1211} \over {22}}} & {{{- 773} \over {20}}} & {{{- 335} \over {36}}} & 0 & 0 & 0 & 0 & 0 & 0 \cr}} \right],\overline {{{\phi ''}_9}} (x) = \left[{\matrix{0 \cr 0 \cr 0 \cr 0 \cr 0 \cr 0 \cr {{1 \over {32256}}{\phi_{1,9}}(x)} \cr {{1 \over {46080}}{\phi_{1,10}}(x)} \cr {{1 \over {63360}}{\phi_{1,11}}(x)} \cr}} \right].

The semi-infinite interval in [0, ∞) in (9–11) can be transformed to [0, 1] by using the coordinate transformation χ=ηη∞ \chi = {\eta \over {{\eta_\infty}}} and the following change of variables (13) F(χ)=f(η)η∞, Θ(χ)=θ(η)η∞,∅ (χ)=φ(η)η∞, F\left( \chi \right) = {{f\left( \eta \right)} \over {{\eta_\infty}}},\;\;\Theta \left( \chi \right) = {{\theta \left( \eta \right)} \over {{\eta_\infty}}},\emptyset \;\left( \chi \right) = {{\varphi \left( \eta \right)} \over {{\eta_\infty}}}, where η^∞ is an unknown finite boundary. Using the above transformations, (9–11) becomes (14) (1+1β)F′′′(χ)+η∞2F(χ)F′′(χ)−2η∞2F′2(χ)−η∞2MF′(χ)=0, \left( {1 + {1 \over \beta}} \right)F'''\left( \chi \right) + \eta_\infty^2F\left( \chi \right)F''\left( \chi \right) - 2\eta_\infty^2{F'^2}\left( \chi \right) - \eta_\infty^2MF'\left( \chi \right) = 0, (15) (1+43R)Θ′′(χ)+Prη∞2F(χ)Θ′(χ)−Prη∞2F′(χ)Θ(χ)+PrHsη∞2Θ(χ)=0, \left( {1 + {4 \over 3}R} \right){\Theta^{''}}\left( \chi \right) + \Pr \eta_\infty^2F\left( \chi \right)\Theta '\left( \chi \right) - \Pr \eta_\infty^2F'\left( \chi \right)\Theta \left( \chi \right) + \Pr Hs\eta_\infty^2\Theta \left( \chi \right) = 0, (16) ∅′′(χ)+Scη∞2F(χ)∅′(χ)−Scη∞2F′(χ)∅(χ)−2γη∞2∅(χ)=0. \left\{{\emptyset \;{\rm{''}}\left( \chi \right) + Sc\eta_\infty^2F\left( \chi \right)\emptyset '\left( \chi \right) - Sc\eta_\infty^2F'\left( \chi \right)\emptyset \left( \chi \right) - 2\gamma \eta_\infty^2\emptyset \left( \chi \right) = 0.} \right\}

The transformed boundary conditions are, (17) F(0)=Sη∞, F′(0)=1,F′(1)=0; Θ(0)=∅ (0)=1η∞, Θ(1)=∅ (1)=0. F\left( 0 \right) = {S \over {{\eta_\infty}}},\;F'\left( 0 \right) = 1,F'\left( 1 \right) = 0;\;\;\Theta \left( 0 \right) = \emptyset \;\left( 0 \right) = {1 \over {{\eta_\infty}}},\;\Theta \left( 1 \right) = \emptyset \;\left( 1 \right) = 0.

The HWM is applied to study the thermohaline convection in MHD Casson fluid over an exponentially stretching sheet. The heat transfer coefficient —θ′(0) results obtained from HWM are validated by comparing with other published studies Pramanik [55]. As shown in Table 2, an excellent agreement is observed. This comparison confirms the suitability of HWM over other numerical techniques for solving highly nonlinear coupled differential equations modeling such complex flow problems.

The fluid velocity decreases with increasing stress parameter as shown in Figures 2 and 4. Since stress is high, the fluid will resist deformation and flow less eagerly. Figure 3 shows that the velocity decreases with increase in the values of the suction/blowing parameter is a result of the interaction between the fluid flow and the applied suction or blowing mechanism. When a suction parameter is applied, it creates a negative pressure gradient in the direction of the flow. This means that the pressure decreases in the direction of the flow. As a result, the fluid is drawn into the system, and the velocity increases. However, as the suction parameter increases beyond a certain point, the negative pressure gradient becomes more significant. This causes a stronger suction effect, leading to a larger draw of fluid into the system. Consequently, the velocity profiles start to decrease because more fluid is being drawn in, reducing the overall velocity within the system. In the case of blowing, a positive pressure gradient is applied in the direction of the flow. This means that the pressure increases in the direction of the flow. Initially, the blowing parameter causes an increase in pressure, which results in an increase in velocity as fluid is pushed through the system. However, as the blowing parameter continues to increase, the positive pressure gradient becomes more substantial. This increased pressure opposes the flow of fluid, leading to a reduction in velocity. The stronger the blowing parameter, the greater the opposition to the fluid flow, causing a decrease in velocity profiles.

Fig. 2

Influence of β on f^′(η).

Fig. 3

Influence of S on f^′(η).

Fig. 4

Influence of β on f(η).

In MHD flow, the interaction between the fluid and the magnetic field can lead to changes in the fluid velocity. The specific effect of the magnetic flux on the velocity of the fluid depends on the direction and strength of the magnetic field, as well as the conductivity of the fluid. In general, when the magnetic parameter increases, which typically refers to an increase in the strength of the magnetic field, the velocity of the fluid tends to decrease. This phenomenon is known as magnetohydrodynamic drag force. The decrease in fluid velocity can be attributed to the Lorentz force, which arises due to the interaction between the magnetic field and the moving charged particles in the fluid. The Lorentz force acts perpendicular to both the magnetic field and the direction of fluid flow. As a result, it can oppose the fluid motion and lead to a decrease in velocity. Furthermore, the Lorentz force can also induce fluid currents that circulate in specific patterns, which can further affect the fluid's velocity. These induced currents can create additional resistance to the fluid flow, causing a decrease in velocity as we can observe in Figures 5 and 6.

Fig. 5

Influence of M on f(η).

Fig. 6

Influence of M on f^′(η).

In the context of temperature profile in Figure 7, increasing the Casson fluid parameter can lead to an increase in the temperature of the fluid. This is because the higher yield stress or plastic viscosity results in a more resistant or sluggish flow behavior. As the fluid requires a greater force to flow, more energy is dissipated as heat due to the increased resistance and interaction between the fluid particles. This dissipated heat can cause an increase in the temperature of the fluid, leading to an increase in the temperature profile. When the suction/blowing parameter increases, it often leads to decreased temperature gradients as shown in Figure 8. In the case of a heat source/sink in a fluid system, the heat added or removed affects the internal energy of the fluid. When the temperature of the fluid increases, it implies that more heat is being added to the system, resulting in an increase in the heat source/sink parameter (Figure 9). The magnetic fluid properties can be influenced by temperature due to the effect of thermal energy on the fluid's molecular structure. When the temperature of a fluid increases seen in Figure 10, the thermal energy causes the molecules within the fluid to move more vigorously, leading to an increase in their kinetic energy. This increased molecular motion can affect the alignment and orientation of the fluid's magnetic dipoles or particles. The temperature profile in a fluid can be influenced by radiative heat transfer. Radiative heat transfer occurs when energy is transferred between objects or regions through electromagnetic waves, without the need for direct physical contact. If there are temperature differences within the fluid, radiation can transfer heat from hotter regions to cooler regions, leading to an increase in the temperature profile (Figure 11). At higher Prandtl numbers, the thermal diffusivity becomes relatively smaller compared to the momentum diffusivity. This means that heat is transported less efficiently compared to momentum in the fluid. Consequently, the convective heat transfer becomes less dominant, and the temperature profiles tend to decrease (Figure 12).

Fig. 7

Influence of β on θ(η).

Fig. 8

Influence of S on θ(η).

Fig. 9

Influence of Hs on θ(η).

Fig. 10

Influence of M on θ(η).

Fig. 11

Influence of R on θ(η).

Fig. 12

Influence of Pr on θ(η).

An increasing Schmidt number can lead to a decreasing concentration profile (Figure 13) due to the dominance of momentum transport over mass transport. The enhanced momentum diffusivity allows for more efficient mixing and dispersion of the fluid, which results in a reduction in the concentration gradients. As the chemical reaction parameter increases, the reaction becomes more intense, leading to a more efficient consumption or production of the species involved. This enhanced reaction activity tends to smoothen out concentration gradients and results in a decreasing concentration profile (Figure 14). The Casson fluid and magnetic field parameters effectively reduce the fluid motion, causing the concentration to increase across the boundary layer for improved mass transfer as shown in Figures 15 and 16.

Fig. 13

Influence of Sc on φ(η).

Fig. 14

Influence of γ on φ(η).

Fig. 15

Influence of β on φ(η).

Fig. 16

Influence of M on φ(η).

The paper applies Hermite wavelet numerical method to solve the governing equations of modeling MHD double diffusive convection in a non-Newtonian fluid (Casson fluid) over an exponentially stretching sheet. The HWM is utilized to handle the highly nonlinear differential equations characterizing the electrically conducting, non-uniform heat source/sink and exponentially stretching sheet problem. We solved the problem by using Mathematica 13.0 version on a 2 G.B. ram sized desktop. Results demonstrated the capability of HWM to accurately solve this complex flow scenario, with solutions closely matching numerical data.

The noteworthy findings obtained from this research can be summarized as follows:

The fluid velocity decreases with increase in the strength of the magnetic field.

Future work will consider a complex non-Newtonian fluid flow problem analysed in computational fluid dynamics.