On the rate of convergence of Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays and their confidence interval estimations

Masataka Hashimoto; Hiroshi Takahashi; Masataka Hashimoto; Hiroshi Takahashi

doi:10.3934/math.2023698

AIMS Mathematics

2023, Volume 8, Issue 6: 13747-13763. doi: 10.3934/math.2023698

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Research article

On the rate of convergence of Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays and their confidence interval estimations

Masataka Hashimoto ¹,
Hiroshi Takahashi ^{2
,
,}

1.
Department of Mathematics, Tokyo Gakugei University, Koganei, 184-8501, Tokyo, Japan
2.
Faculty of Business and Commerce, Keio University, Yokohama, 223-8521, Kanagawa, Japan

Received: 27 October 2022 Revised: 10 March 2023 Accepted: 03 April 2023 Published: 11 April 2023
MSC : 60H10, 65U05

In this paper, we investigate Euler–Maruyama approximate solutions of stochastic differential equations (SDEs) with multiple delay functions. Stochastic differential delay equations (SDDEs) are generalizations of SDEs. Solutions of SDDEs are influenced by both the present and past states. Because these solutions may include past information, they are not necessarily Markov processes. This makes representations of solutions complicated; therefore, approximate solutions are practical. We estimate the rate of convergence of approximate solutions of SDDEs to the exact solutions in the $L^p$ -mean for $p \geq 2$ and apply the result to obtain confidence interval estimations for the approximate solutions.

Keywords:

Citation: Masataka Hashimoto, Hiroshi Takahashi. On the rate of convergence of Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays and their confidence interval estimations[J]. AIMS Mathematics, 2023, 8(6): 13747-13763. doi: 10.3934/math.2023698

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Abstract

List of symbols:

$W, V, U$	Velocity components ( $m{s}^{-1}$ )	$y, x, z$	Space coordinates ( $m$ )
$\nu$	Kinematic viscosity $\left({m}^{2}{s}^{-1}\right)$	$\beta$	Casson fluid number
$\rho$	Fluidic density $\left(Kg{m}^{-3}\right)$	${\beta }_{i}, {\beta }_{e}$	Ion slip and Hall parameters
$T$	Temperature $\left(K\right)$	${T}_{\infty }$	Ambient temperature $\left(K\right)$
$\mu$	Viscosity $\left(Kg{m}^{-1}{s}^{-1}\right)$	$G$	Gravitational acceleration $\left(N\right)$
$\beta, \gamma$	Buoyancy parameters	${B}_{0}$	Magnetic induction $\left(A.Kg{s}^{-2}\right)$
$\sigma$	Electrical conductivity $\left(S{m}^{-1}\right)$	$n$	Power law index number
$a, b$	Constants	${T}_{w}$	Wall temperature $\left(K\right)$
$\eta$	Independent variable	$Pr$	Prandtl number
$M$	Magnetic number	${\phi }_{3}, {\phi }_{1}, {\phi }_{2}$	Volume fractions
$\lambda$	Stretching ratio number	$Cf$	Skin friction coefficient
$Nu$	Nusselt number	$TH$	Tri-hybrid nanofluid
$bf$	Base fluid	$EG$	Ethylene glycol
${Al}_{2}{O}_{3}$	Aluminum oxide	$Si{O}_{2}$	Silicon dioxide
${w}_{1}, {w}_{3}, {w}_{2}, {w}_{4}, {w}_{5}$	Weight functions	$\infty$	Infinity
$K$	Thermal conductivity $\left(W{m}^{-1}\right)$	PDEs	Partial differential equations
ODEs	Ordinary differential equations	${sp}_{1}, {sp}_{3}, {sp}_{2}$	Solid nanoparticles
${\theta }_{\omega }$	Temperature ratio number	${N}_{r}$	Thermal radiation number
$\epsilon$	Porosity parameter $\left({m}^{2}\right)$	${f}_{s}$	Inertia coefficient
${k}^{1}$	Permeability of porous $\left({m}^{2}\right)$	${F}_{r}$	Forchheirmer number
${\sigma }^{}, {k}^{}$	Stefan-Boltzmann constants	$\alpha$	Thermal diffusivity $\left({m}^{2}{s}^{-1}\right)$

1. Introduction

Nanofluid is an engineered colloidal mixture regarding base fluid with single nanoparticles. Such behavior of fluid is applicable in cooling process, thermal process, thermal enhancement, thermal reduction and microelectronics. Deep exploration regarding case of mixture of base fluid with two kind's nanoparticles is termed as hybrid nanofluid. Moreover, Casson fluid behaves like a non-Newtonian liquid and shear thinning. Such type of non-Newtonian rheology is applicable in petroleum products, syrup drugs, production of plastic materials and engineering field etc. Rheology related to Casson fluid was investigated by Casson ^[1]. He has investigated that Casson model displays role like a shear thinning, plastic liquid model and high shear viscosity. Chabani et al. ^[2] discussed features of hybrid nanoparticles involving Lorentz force in triangular enclosure. They have studied convective flow considering the Darcy Forchheirmer model and included that flow is reduced when Lorentz force is implemented. Chu et al. ^[3] investigated a study related to hybrid nanofluid in the presence of magnetohydrodynamics for unsteady flow in two parallel plates including shape effects. They considered engine oil as base fluid and model is numerical solved by numerical scheme. Selimefendigil et al. ^[4] studied the role of various shapes regarding nanoparticles in ventilated cavities considering artificial neural networks. They implemented a finite element approach to find numerical results. It was included that maximum temperature was obtained using L-shaped cavity. Saleem et al. ^[5] estimated performance of several shapes of nanoparticles in horizontal surface considering base fluid (water). They have used a shooting approach to find parametric study on temperature and flow profiles. It was established that mass transport rate is enhanced versus erratic motion. Saleem et al. ^[6] established modeling related to Eyring-Powell material considering magnetized effect in mass species and heat transfer involving chemical reaction using finite element approach. Algehyne et al. ^[7] studied thermal features of Maxwell fluid in hybrid particles over a stretching frame. They have utilized finite element approach to find computational study and parametric study. It was investigated that highest thermal performance can be achieved by hybrid nanofluid rather than nanofluid. Imran et al. ^[8] performed thermal aspects of the solar collector using approach of hybrid nanoparticles via several shape effects implementing a numerical approach. Several studies regarding hybrid nanofluid are mentioned in ^[9]. Khan et al. ^[10] discussed features of Lorentz force in Casson fluid inserting dust particles along with hybrid nanoparticles past a stretchable sheet using an analytic approach.

Ternary hybrid nanofluid is a mixture of three kinds of nanoparticles in base fluid. It is an efficient approach to achieve the highest temperature rather than nanofluid and hybrid nanofluid. The process regarding tri-hybrid nanoparticles is most efficient for development of thermal enhancement and thermal reduction. Sarada et a. ^[11] estimated enhancement process in energy efficient using applications of ternary hybrid nanoparticles in curved stretching sheet implementing shooting approach. Nazir et al. ^[12] visualized comparison enhancement into different fluids using Hall currents inserting mixture of multiple nanoparticles in base fluid via finite element methodology. Dezfulizadeh et al. ^[13] discussed impacts of Lorentz force in MHD flow using exergy an efficiency approach in the presence of ternary hybrid nanofluid. Oke ^[14] determined performance of tri-hybrid nanofluid in mass dissuasion and thermal energy considering base fluid (EG) past a rotating surface. Xiu et al. ^[15] studied dynamic behavior of ternary-hybrid nanoparticles in mass diffusion and thermal field considering small and large volume of nanoparticles with platelet, spherical and cylindrical nanoparticles over wedge using shooting approach. They have found that tri-hybrid nanofluid was observed very significantly for thermal performance rather than hybrid nanofluid. Animasaun and Asogwa ^[16] discussed comparative influences among alumina and water-based nanoparticles including based cupric nanoparticles (water-based) over stretching surface. Rasool et al. ^[17] experienced multiple aspects based on radiation and viscous dissipation using theory of non-Fourier's in second grade rheology using analytic approach on porous surface. Ashraf et al. ^[18] captured thermal features of magnetohydrodynamic flow in peristaltic fluid inserting magnetite nanoparticles in blood using the D-quadrature algorithm.

Nowadays, several industrial applications in pigments, electronics, cosmetics, food processing and utilizing engineering process metal oxide nanomaterials such as silicon dioxide, aluminum oxide and titanium dioxide. The retention and transport of nanoparticles $\left({Al}_{2}{O}_{3}, Si{O}_{2}, Ti{O}_{2}\right)$ ware utilized to resolve issues of petroleum engineering and environmental. Nanoparticles $\left({Al}_{2}{O}_{3}, Si{O}_{2}, Ti{O}_{2}\right)$ were introduced for enhancement of oil recovery by Hendraningrat et al. ^[19]. ${Al}_{2}{O}_{3}$ nanoparticles have higher boiling and melting points which are useful for enhancement of thermal inertia (see Song et al. ^[20]). In addition, ${Al}_{2}{O}_{3}$ nanoparticles are applicable to enhance the strengthens of smoothness, ceramics, creep resistance and crack hardness. Dynamic features of thermal transport in the presence of partially ionized liquid subjected to magnetic field is significantly exaggerated by magnetic field. Animasaun et al. ^[21] investigated that the Hall effect occurs when a magnetic field is applied perpendicular to the current flow through a thin sheet. As a result, an electric field is created parallel to the magnetic and current fields and directly proportional to the product of the magnetic induction and current density. Animasaun et al. ^[22] discussed comparison impacts among 47 nm and 36 nm alumina nanoparticles in base fluid (water) under action of Hall currents. Farooq et al. ^[23] developed model of entropy production rate in mass diffusion and thermal field under effects of Hall currents, viscous dissipation, magnetic field and Joule heating in cylindrical tubules.

The purpose of the present flow problem is to study characterizations of Casson fluid involving Lorentz force and thermal radiation (non-linear) over a 3D vertical surface accompanied with buoyancy force, Forchheirmer porous model and Joule heating. In additionally, mixture of three kinds of nanoparticles ( ${Al}_{2}{O}_{3}, Si{O}_{2}, Ti{O}_{2}$ ) in ethylene glycol is inserted to visualize thermal performance among cylindrical and platelet nanoparticles. The current problem fills the gap while not adequately studied with the following points.

● Analysis of heat transfer in Casson liquid under effects of Joule heating and viscous dissipation, thermal radiation (non-linear), Hall and ion slip forces using Darcy's Forchheirmer theory past a 3D surface has not been investigated yet.

● Previous to this investigation, no comparative analysis on Casson fluid among cylindrical and platelet nanoparticles over 3D surface using Forchheirmer porous model has been existed in literature works computed by finite element method.

● In existence literature, no investigations have been reported including ternary hybrid nanoparticles and shape effects rather than ^[24] to visualize comparative analysis among cylindrical and platelet nanoparticles.

Existence works demonstrate that development of Casson liquid inserting ternary hybrid nanofluid using thermal properties of various shapes of nanoparticles over 3-dimensional vertical using Forchheirmer porous model frame is not studied yet. Moreover, ion slip and Hall currents are considered with viscous dissipation and variable thermal conductivity. Present complex development is numerically handled by a finite element approach. In addition, section Ⅰ, section Ⅱ, section Ⅳ and section Ⅴ are based on literature review, problem modeling, numerical approach, results and conclusions, respectively.

2. Modeling of developed model

A 3D model of Casson fluid is developed over a stretching surface using theories of generalized Ohm's law and Darcy's Forchheirmer model. Three types of nanoparticles are inserted into EG (ethylene glycol) while various shapes of nanoparticles (cylindrical and platelet) are addressed. Motion of nanoparticles is produced using movement of wall velocity. Buoyancy force is considered whereas viscous dissipation and thermal radiation (non-linear) are added in the energy equation. Suspension of nanoparticles regarding aluminum oxide, titanium oxide and silicon dioxide in working fluid is considered. Reduced form regarding Navier-Stoke equations is obtained and physical model is considered by . Thermal radiation in terms of non-linear is utilized to determine thermal fields. Surface is assumed as non-conducting in terms of electrically and non-isothermal. is 3D vertical surface in $y$ -, $x$ - and z-components while non-uniform magnetic field is considered along z-component and gravitational force acts along downward direction. Partially flow over surface is induced by bidirectional movement of wall velocities ${U}_{w}$ and ${V}_{w}$ at $z = 0.$ A 3D surface lies in plane at $z = 0$ and flow occupies region in field at $z > 0.$ Wall temperature and ambient temperature are considered as ${T}_{w}$ and ${T}_{\mathrm{\infty }}.$ Further, it is studied that ethylene glycol is shear thinning ^[25], non-Newtonian liquid ^[26] and plasma ^[26]. Casson fluid model is also known as non-Newtonian liquid and shear thinning ^[1]. Table 1 reveals thermal properties of various nanoparticles. So, its rheology is prescribed by the Casson fluid stress tensor relation.

Figure 1. Physical representation of 3D model.

DownLoad: Full-Size Img PowerPoint

Table 1. Thermal conductivity, density and electrical conductivity in EG ^[27].

	$k$	$\rho$	$\sigma$
${Al}_{2}{O}_{3}$	32.9	6310	$5.96\times {10}^{7}$
$Si{O}_{2}$	1.4013	2270	$3.5\times {10}^{6}$
$Ti{O}_{2}$	8.953	4250	$2.4\times {10}^{6}$
EG	0.144	884	$0.125\times {10}^{-11}$

| Show Table

DownLoad: CSV

Conservation laws for energy equation, momentum equation for incompressible and steady flow ^[24] using Casson liquid are

${U}_{x}+{V}_{y}+{W}_{z} = 0,$

(1)

$\begin{array}{c} U{U}_{x}+V{U}_{y}+W{U}_{z} = \frac{{\mu }_{TH}}{{\rho }_{TH}}\left(1+\frac{1}{\beta }\right){U}_{zz}-\frac{{\mu }_{TH}}{{\rho }_{TH}}{f}_{s}U-\frac{{f}_{s}}{{\left({k}^{1}\right)}^{1/2}}{U}^{2} \\ +\frac{{{\left({B}_{0}\right)}^{2}{\left(y+x\right)}^{n-1}\sigma }_{TH}}{{\rho }_{TH}\left[{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\beta }_{e}^{2}\right]}\left[{\beta }_{e}V-\left(1+{\beta }_{e}{\beta }_{i}\right)U\right]+G{\beta }_{TH}\left(T-{T}_{\infty }\right), \end{array}$

(2)

$\begin{array}{c} U{V}_{x}+V{V}_{y}+W{V}_{z} = \frac{{\mu }_{TH}}{{\rho }_{TH}}\left(1+\frac{1}{\beta }\right){V}_{zz}-\frac{{\mu }_{TH}}{{\rho }_{TH}}{f}_{s}V-\frac{{f}_{s}}{{\left({k}^{1}\right)}^{1/2}}{V}^{2} \\ -\frac{{{\left({B}_{0}\right)}^{2}{\left(y+x\right)}^{n-1}\sigma }_{TH}}{{\rho }_{TH}\left[{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\beta }_{e}^{2}\right]}\left[{\beta }_{e}U+\left(1+{\beta }_{e}{\beta }_{i}\right)V\right]+G{\gamma }_{TH}\left(T-{T}_{\infty }\right), \end{array}$

(3)

$\begin{array}{c} U{T}_{x}+V{T}_{y}+W{T}_{z} = \frac{{{\left({B}_{0}\right)}^{2}{\left(y+x\right)}^{n-1}\sigma }_{TH}}{{\rho }_{TH}\left[{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\beta }_{e}^{2}\right]}\left({U}^{2}+{V}^{2}\right)+{\left(\alpha {T}_{z}+\frac{16{T}^{3}{\sigma }^{*}}{3{k}^{*}{\left(\rho {C}_{p}\right)}_{f}}{T}_{z}\right)}_{z}\\ +\frac{{\mu }_{TH}}{{\left(\rho {C}_{p}\right)}_{TH}}\left(1+\frac{1}{\beta }\right)\left[{\left({U}_{z}\right)}^{2}+{\left({V}_{z}\right)}^{2}\right], \end{array}$

(4)

Boundary conditions ^[24] of developing model are defined as

$\begin{array}{c} U = a{\left(x+y\right)}^{n}, V = {b\left(x+y\right)}^{n}, W = 0, asz = 0, U = 0, \\ T\to {T}_{\infty }, V = 0, asz\to \infty , T = {T}_{w} = {T}_{\infty }+{A}_{1}{T}_{0}{\left(y+x\right)}^{2n}. \end{array}$

(5)

Similarity variables ^[24] are delivered as

$\begin{array}{c} U = a{\left(y+x\right)}^{n}{F}^{{'}}, V = b{\left(y+x\right)}^{n}{G}^{{'}}, \eta = \sqrt{\frac{a}{{\nu }_{f}}}{\left(y+x\right)}^{\frac{n-1}{2}}z,\\ W = -\sqrt{a{\nu }_{f}}{\left(x+y\right)}^{\frac{n-1}{2}}\left\{\frac{n+1}{2}\left(F+G\right)+\frac{n-1}{2}\eta \left({F}^{{'}}+{G}^{{'}}\right)\right\}. \end{array}$

(6)

Equations (1)–(4) into dimensionless form ^[24] using variables similarity are

$\begin{array}{c} \left(1+\frac{1}{\beta }\right){F}^{{'}{'}{'}}-{A}_{1}\left[-n\left({F}^{{'}}+{G}^{{'}}\right){F}^{{'}}+\frac{n+1}{2}\left(F+G\right){F}^{{'}{'}}\right]-\epsilon {F}^{{'}}-\frac{{\nu }_{f}}{{\nu }_{TH}}Fr{F{'}}^{2} \\ +\frac{{M}^{2}{\left(1-{\phi }_{1}\right)}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}}{{\left(1-{\phi }_{3}\right)}^{-2.5}\left[{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}\right]}\left[{\beta }_{e}{G}^{{'}}-\left(1+{\beta }_{e}{\beta }_{i}\right){F}^{{'}}\right]+{\alpha }_{1}\theta = 0, \end{array}$

(7)

$\begin{array}{c} \left(1+\frac{1}{\beta }\right){G}^{{'}{'}{'}}+{A}_{1}\left[n\left({F}^{{'}}+{G}^{{'}}\right){G}^{{'}}-\frac{n+1}{2}\left(F+G\right){G}^{{'}{'}}\right]-\epsilon {G}^{{'}}-\frac{{\nu }_{f}}{{\nu }_{TH}}Fr{G{'}}^{2}\\ -\frac{{M}^{2}{\left(1-{\phi }_{1}\right)}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}}{{\left(1-{\phi }_{3}\right)}^{-2.5}\left[{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}\right]}\left[{\beta }_{e}{F}^{{'}}+\left(1+{\beta }_{e}{\beta }_{i}\right){G}^{{'}}\right]+{\alpha }_{2}\theta = 0, \end{array}$

(8)

$\begin{array}{c} {\left(\left(1+{N}_{r}{\left(1+\left({\theta }_{\omega }-1\right)\theta {'}\right)}^{3}\right)\right)}^{{'}}+\frac{{k}_{f}{\left(\rho {c}_{p}\right)}_{TH}}{{k}_{TH}{\left(\rho {c}_{p}\right)}_{f}}Pr\left[\frac{n+1}{2}{\theta }^{{'}}\left(F+G\right)-n\theta \left({F}^{{'}}+{G}^{{'}}\right)\right] \\ +\frac{{k}_{f}}{{k}_{TH}}\frac{PrEc{M}^{2}}{{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}}\left[{\left({F}^{{'}}\right)}^{2}+{\left({G}^{{'}}\right)}^{2}\right] = 0, \end{array}$

(9)

BCs in term of dimensionless ^[24] are

$F\left(0\right) = {F}^{{'}}\left(\infty \right) = 0, {F}^{{'}}\left(0\right) = 1, \left(0\right) = {G}^{{'}}\left(\infty \right) = 0, {G}^{{'}}\left(0\right) = \lambda , \theta \left(0\right) = 1 = 0, \theta \left(\infty \right) = 0.$

(10)

Correlations regarding ternary hybrid nanofluid ^[28] for density, thermal conductivity and viscosity are defined as

${A}_{1} = \left(1-{\phi }_{1}-{\phi }_{2}-{\phi }_{3}\right)+{\phi }_{1}\frac{{\rho }_{s{p}_{1}}}{{\rho }_{f}}+{\phi }_{2}\frac{{\rho }_{s{p}_{2}}}{{\rho }_{f}}+{\phi }_{3}\frac{{\rho }_{s{p}_{3}}}{{\rho }_{f}},$

(11)

${\mu }_{TH} = {\mu }_{Nf1}{\phi }_{1}+{\mu }_{Nf2}{\phi }_{2}+{\mu }_{Nf3}{\phi }_{3}, {K}_{TH} = \frac{{K}_{Nf1}{\phi }_{1}+{K}_{Nf2}{\phi }_{2}+{K}_{Nf3}{\phi }_{3}}{\phi },$

(12)

${\rho }_{TH} = \left(1-{\phi }_{1}-{\phi }_{2}-{\phi }_{3}\right){\rho }_{bf}+{\phi }_{1}{\rho }_{s{p}_{1}}+{\phi }_{2}{\rho }_{s{p}_{2}}+{\phi }_{2}{\rho }_{s{p}_{2}},$

(13)

${\left(\rho {C}_{p}\right)}_{TH} = \left(1-{\phi }_{1}-{\phi }_{2}-{\phi }_{3}\right){\left(\rho {C}_{p}\right)}_{bf}+{\phi }_{1}{\left(\rho {C}_{p}\right)}_{s{p}_{1}}+{\phi }_{2}{\left(\rho {C}_{p}\right)}_{s{p}_{2}}+{\phi }_{3}{\left(\rho {C}_{p}\right)}_{s{p}_{3}},$

(14)

Thermal conductivity and viscosity models for cylindrical and platelet (see Figure 2) ^[28] are defined as

$\frac{{\mu }_{Nf2}}{{\mu }_{bf}} = 1+904.4{\phi }^{2}+13.5\phi , {K}_{nf1} = {K}_{bf}\left[\frac{{K}_{s{p}_{2}}+3.9{K}_{bf}-3.9\phi \left({K}_{bf}-{K}_{s{p}_{2}}\right)}{{K}_{s{p}_{2}}+3.9{K}_{bf}+\phi \left({K}_{bf}-{K}_{s{p}_{2}}\right)}\right],$

(15)

$\frac{{\mu }_{Nf3}}{{\mu }_{bf}} = 1+612.6{\phi }^{2}+37.1\phi , {K}_{nf1} = {K}_{bf}\left[\frac{{K}_{s{p}_{3}}+4.7{K}_{bf}-4.7\phi \left({K}_{bf}-{K}_{s{p}_{3}}\right)}{{K}_{s{p}_{3}}+4.7{K}_{bf}+\phi \left({K}_{bf}-{K}_{s{p}_{3}}\right)}\right].$

(16)

Figure 2. Representation of platelet and cylindrical nanoparticles.

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Wall shear stresses due to ternary hybrid nanofluid ^[24] are derived as

${Cf = \frac{{\left.{\tau }_{zx}\right|}_{z = 0}}{{\rho }_{TH}{\left({u}_{w}\right)}^{2}}, Re}^{1/2}Cf = \frac{{\left(1+{\phi }_{3}\right)}^{-2.5}{\left(n+1\right)}^{1/2}}{{\left(1+{\phi }_{2}\right)}^{2.5}{\left(1+{\phi }_{1}\right)}^{2.5}}\left(1+\frac{1}{\beta }\right){F}^{{'}{'}}\left(0\right),$

(17)

${Cg = \frac{{\left.{\tau }_{zy}\right|}_{z = 0}}{{\rho }_{TH}{\left({u}_{w}\right)}^{2}}, Re}^{1/2}Cg = \frac{{\left(1+{\phi }_{3}\right)}^{-2.5}{\left(n+1\right)}^{1/2}}{{\left(1+{\phi }_{2}\right)}^{2.5}{\left(1+{\phi }_{1}\right)}^{2.5}}\left(1+\frac{1}{\beta }\right){G}^{{'}{'}}\left(0\right).$

(18)

Temperature gradient due to ternary hybrid nanofluid ^[24] is

$Nu = \frac{-{\left.{K}_{TH}\left(y+x\right){T}_{z}\right|}_{z = 0}}{\left({T}_{w}-{T}_{\infty }\right){K}_{f}}, Nu{Re}^{-1/2}-\frac{{K}_{TH}}{{K}_{f}}{\theta }^{{'}}\left(0\right).$

(19)

3. Computational approach

Finite element method ^[6,7] is employed to construct numerical solution of desired problem. It is described under.

Step 1: Firstly, the problem domain is transformed into a finite number of elements and derived linear type polynomial over each element. Weak form can be achieved from strong form using weighted residual function. Weighted residuals ^[6,7] are derived as

${\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{w}_{1}\left[{F}^{{'}}-H\right]d\eta = 0, {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{w}_{2}\left[{G}^{{'}}-T\right]d\eta = 0,$

(20)

${\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{w}_{3}\left[\begin{array}{c}\left(1+\frac{1}{\beta }\right){H}^{{'}{'}}-{A}_{1}\left(-n\left(H+T\right)H+\frac{n+1}{2}\left(F+G\right){H}^{{'}}\right)\\ +\frac{{M}^{2}{\left(1-{\phi }_{1}\right)}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}}{{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}}\left({\beta }_{e}T-\left(1+{\beta }_{e}{\beta }_{i}\right)H\right)+{\alpha }_{1}\theta \end{array}\right]d\eta = 0,$

(21)

${\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{w}_{4}\left[\begin{array}{c}\left(1+\frac{1}{\beta }\right)T{'}{'}+{A}_{1}\left(n\left(H+T\right)T-\frac{n+1}{2}\left(F+G\right)T{'}\right)+{\alpha }_{2}\theta \\ +\frac{{M}^{2}{\left(1-{\phi }_{1}\right)}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}}{{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}}\left({\beta }_{e}H+\left(1+{\beta }_{e}{\beta }_{i}\right)T\right)\end{array}\right]d\eta = 0,$

(22)

${\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{w}_{5}\left[\begin{array}{c}\theta {'}{'}+\frac{{k}_{f}}{{k}_{TH}}\frac{PrEc{M}^{2}}{{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}}\left\{{\left(H\right)}^{2}+{\left(H\right)}^{2}\right\}\\ -\frac{{k}_{f}{\left(\rho {c}_{p}\right)}_{TH}}{{k}_{TH}{\left(\rho {c}_{p}\right)}_{f}}Pr\left\{\frac{n+1}{2}\theta {'}\left(F+G\right)-n\theta \left(H+T\right)\right\}\end{array}\right]d\eta = 0.$

(23)

Here, ${w}_{3}, {w}_{1}, {w}_{2}, {w}_{4}$ and ${w}_{5}$ are weighted functions. Unknown variables and shape functions ^[6] are defined as

$F = \sum _{j = 1}^{2}{F}_{i}{\psi }_{j}, T = \sum _{j = 1}^{2}{T}_{i}{\psi }_{j}, G = \sum _{j = 1}^{2}{G}_{i}{\psi }_{j}, H = \sum _{j = 1}^{2}{H}_{i}{\psi }_{j}, \theta = \sum _{j = 1}^{2}{\theta }_{i}{\psi }_{j},$

(24)

${\psi }_{j} = {\left(-1\right)}^{i-1}\left(\frac{\eta -{\eta }_{j-1}}{{\eta }_{i}-{\eta }_{j-1}}\right).$

(25)

Step 2: Boundary integral vectors, stiffness matrices and force vectors are derived. After it, the global stiffness matrix is derived. In additionally, Picard linearization approach is utilized for achieving linear form of algebraic equations. Derivation of stiffness elements are

${K}_{ij}^{11} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{\psi }_{i}\left(\frac{d{\psi }_{j}}{d\eta }\right)d\eta , {K}_{ij}^{13} = -{\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{\psi }_{i}\left(\frac{d{\psi }_{j}}{d\eta }\right)d\eta , {K}_{ij}^{12} = {K}_{ij}^{14} = {K}_{ij}^{15} = 0,$

(26)

${K}_{ij}^{21} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{\psi }_{i}\left(\frac{d{\psi }_{j}}{d\eta }\right)d\eta , {K}_{ij}^{24} = -{\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{\psi }_{i}\left(\frac{d{\psi }_{j}}{d\eta }\right)d\eta , {K}_{ij}^{22} = {K}_{ij}^{23} = {K}_{ij}^{25} = 0,$

(27)

${K}_{ij}^{33} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}\left[\begin{array}{c}-\left(1+\frac{1}{\beta }\right)\frac{d{\psi }_{i}}{d\eta }\frac{d{\psi }_{j}}{d\eta }-{A}_{1}\left(-n\left(\stackrel{-}{H}+\stackrel{-}{T}\right){\psi }_{i}{\psi }_{j}+\frac{n+1}{2}\left(\stackrel{-}{F}+\stackrel{-}{G}\right){\psi }_{i}{\psi }_{j}\right)\\ -\frac{{M}^{2}{\left(1-{\phi }_{1}\right)}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}}{{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}}\left(\left(1+{\beta }_{e}{\beta }_{i}\right){\psi }_{i}{\psi }_{j}\right)\end{array}\right]d\eta ,$

(28)

${K}_{ij}^{34} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}\left({\beta }_{e}{\psi }_{i}{\psi }_{j}\right)d\eta , {K}_{ij}^{35} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{\alpha }_{1}\left({\psi }_{i}{\psi }_{j}\right)d\eta , {K}_{ij}^{31} = 0, {K}_{ij}^{32} = 0, {B}_{i}^{5} = 0,$

(29)

${K}_{ij}^{34} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}\left[\begin{array}{c}-{A}_{1}\left(n\left(\stackrel{-}{H}+\stackrel{-}{T}\right){\psi }_{i}{\psi }_{j}-\frac{\left(\left(\stackrel{-}{F}+\stackrel{-}{G}\right){\psi }_{i}\frac{d{\psi }_{j}}{d\eta }\right)n+1}{2}\right)\\ -\frac{{M}^{2}{\left(1-{\phi }_{1}\right)}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}}{{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}}\left(\left(1+{\beta }_{e}{\beta }_{i}\right){\psi }_{i}{\psi }_{j}\right)\end{array}\right]d\eta , {B}_{i}^{4} = 0, {K}_{ij}^{42} = 0,$

(30)

${K}_{ij}^{55} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}\left[\begin{array}{c}-{\left(1+{N}_{r}{\left(1+\left({\theta }_{\omega }-1\right)\right)}^{3}\right)}^{{'}}3{\stackrel{-}{\theta {'}}}^{2}\frac{d{\psi }_{i}}{d\eta }\frac{d{\psi }_{j}}{d\eta }-\frac{{k}_{f}{\left(\rho {c}_{p}\right)}_{TH}}{{k}_{TH}{\left(\rho {c}_{p}\right)}_{f}}n\left(\stackrel{-}{H}+\stackrel{-}{T}\right){\psi }_{i}{\psi }_{j}\\ -\frac{{k}_{f}{\left(\rho {c}_{p}\right)}_{TH}}{{k}_{TH}{\left(\rho {c}_{p}\right)}_{f}}Pr\stackrel{-}{G}{\psi }_{i}\frac{d{\psi }_{j}}{d\eta }+\frac{{k}_{f}{\left(\rho {c}_{p}\right)}_{TH}}{{k}_{TH}{\left(\rho {c}_{p}\right)}_{f}}Pr\left(\frac{n+1}{2}\right)\stackrel{-}{F}{\psi }_{i}\frac{d{\psi }_{j}}{d\eta }\end{array}\right]d\eta ,$

(31)

${K}_{ij}^{53} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}\left[\frac{{k}_{f}}{{k}_{TH}}\frac{PrEc{M}^{2}}{{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}}\stackrel{-}{H}{\psi }_{i}{\psi }_{j}\right]d\eta , {K}_{ij}^{41} = {K}_{ij}^{51} = 0, {K}_{ij}^{52} = 0, {B}_{i}^{1} = 0, {K}_{ij}^{43} = 0,$

(32)

${K}_{ij}^{54} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}\left[\frac{{k}_{f}}{{k}_{TH}}\frac{PrEc{M}^{2}}{{\left(1+{\beta }_{i}{\beta }_{e}\right)}^{2}+{\left({\beta }_{e}\right)}^{2}}\stackrel{-}{G}{\psi }_{i}{\psi }_{j}\right]d\eta , {B}_{i}^{2} = 0, {B}_{i}^{3} = 0, {K}_{ij}^{35} = {\int }_{{\eta }_{e}}^{{\eta }_{e+1}}{\alpha }_{1}\left({\psi }_{i}{\psi }_{j}\right)d\eta ,$

(33)

Step 3: System of linear equations is solved in form of iteratively under numerically tolerance ( ${10}^{-5}$ ). Convergence and error analysis is mentioned below

$Err = \left|{\delta }^{i}-{\delta }^{i-1}\right|, Max\left|{\delta }^{i}-{\delta }^{i-1}\right|\le {10}^{-8}.$

(34)

Step 4: MAPLE 18 is used to design programming of finite element approach. Mesh free study is recorded in Table 2. Flow chart of numerical methodology is addressed by Figure 3. Validation results of present model is verified with already published work ^[24] considering absence of ternary hybrid nanofluid, power law index number, Darcy's Forchheirmer model and Casson fluid. Validation of numerical results with published work is mentioned in Table 3.

Table 2. Numerical study of mesh free analysis ^[6] for

$F{'}\left(\frac{{\eta }_{max}}{2}\right)$ ,

$G{'}\left(\frac{{\eta }_{max}}{2}\right)$ and

$\theta \left(\frac{{\eta }_{max}}{2}\right)$ .

$e$	$F{'}\left(\frac{{\eta }_{max}}{2}\right)$	$G{'}\left(\frac{{\eta }_{max}}{2}\right)$	$\theta \left(\frac{{\eta }_{max}}{2}\right)$
30	0.5172614186	0.4141116019	0.5331053851
60	0.5009743321	0.3991196957	0.5166109650
90	0.4955010944	0.3940814743	0.5110865969
120	0.4927563199	0.3915549156	0.5083195950
150	0.4911066420	0.3900362436	0.5066578927
180	0.4900077513	0.3890241819	0.5055494665
210	0.4892201185	0.3882992913	0.5047574611
240	0.4886298879	0.3877558788	0.5041632925
270	0.4881717384	0.3873337477	0.5037009996
300	0.4888027764	0.3879944431	0.5033311740

| Show Table

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Figure 3. Flow diagram of solution approach.

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Table 3. Validation of present problem for wall shear stresses with published study ^[24] considering

$n = 0, {\phi }_{1} = 0, M = 0, {\phi }_{2} = 0,$

${\phi }_{3} = 0, \beta \to \mathrm{\infty }, \epsilon = 0, Fr = 0$ .

$n$	$\lambda$	Khan et al. ^[24]	present results for ${-F}^{{'}{'}}\left(0\right)$	Khan et al. ^[24]	present results for $-{G}^{{'}{'}}\left(0\right)$
	0.0	1.0	1.0	0.0	0.0
1	0.5	1.224745	0.2210013530	0.612372	0.6124360703
	1.0	1.414214	0.4111910216	1.414214	0.4145020304
	0.0	1.624356	0.6238906653	0.0	0.0
3	0.5	1.989422	0.9893026033	0.994711	0.9950661609
	1.0	2.297182	0.2969018879	2.297182	2.2960670811

| Show Table

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4. Results and discussion consequences

The parametric study is captured to determine flow behavior and transport of thermal energy into a mixture of ternary hybrid nanoparticles in base and working fluids. The graphical outcomes are displayed in –. It is noticed that ternary hybrid nanofluid is a mixture of aluminum dioxide, titanium dioxide, silicon dioxide in base fluid and working fluid. Hybrid nanofluid is considered a mixture of silicon dioxide, titanium dioxide while nanofluid is a mixture of titanium dioxide in Casson fluid. In additionally, thermal and velocity fields are plotted versus different parameters implementing ranges of $0.5\le \beta \le 4,$ $0.1\le n\le 3$ , $0.0\le M\le 2.0$ , $, 0.0\le {\beta }_{e}\le 3.0$ , $, 0.0\le {\beta }_{i}\le 4.0$ , $, 204\le Pr\le 208$ , $0.0\le Ec\le 3.0$ , $0.0\le {\alpha }_{1}\le 1.5$ , $0.0\le {\phi }_{1}\le 0.04$ , $0.0\le \lambda \le 0.5$ , $\text{0.0≤}{\phi }_{2}\le 0.07$ , $0.0\le {\phi }_{2}\le 0.07$ , $0.0\le {\phi }_{3}\le 0.06$ , $0.0\le {\alpha }_{1}\le 2.0$ , $0.0\le \epsilon \le 1.5$ , $0.0\le Fr\le 3.0$ , $0.0\le {N}_{r}\le 3.0$ and $0.0\le {\theta }_{\omega }\le 2.0.$

Figure 4. Distribution in velocity fields with

$\beta$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. Distribution in velocity fields with

${\beta }_{e}$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. Distribution in velocity fields with

${\beta }_{i}$ .

DownLoad: Full-Size Img PowerPoint

Figure 7. Distribution in velocity fields with

$Fr$ .

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Figure 8. Distribution in thermal field with

${N}_{r}$ .

DownLoad: Full-Size Img PowerPoint

Figure 9. Distribution in thermal field with

$Fr$ .

DownLoad: Full-Size Img PowerPoint

Figure 10. Distribution in thermal field with

${\beta }_{e}$ .

DownLoad: Full-Size Img PowerPoint

Figure 11. Distribution in thermal field with

${\beta }_{i}$ .

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Figure 12. Comparison influences among fluid, hybrid nanofluid, trihybrid nanofluid and nanofluid.

DownLoad: Full-Size Img PowerPoint

Figures 4– are plotted to determine the effect of several parameters on flow involving dispersion of ternary hybrid nanoparticles. It is visualized that platelet nanoparticles are represented by solid curves on graphs and dot curves are associated with cylindrical nanoparticles on graphs. clearly reveal that there is reduction into flow (y- and x-directions) when $\beta$ is changed. The effect of the Casson parameter for cylindrical nanoparticles is less than the effect of the Casson number for platelet nanoparticles. This effect is created using concept of Casson fluid in momentum equations whereas existence of $\beta$ is formulated considering tensor of Casson fluid in momentum equations. Flow becomes reduced in y- and x-directions because of $\beta$ is appeared in form of $\frac{1}{\beta }$ . Therefore, an inverse proportional relation can be investigated versus velocity fields. Consequently, flow slows down in terms of y- and x-directions. Casson fluid behaves as a shear thinning and non-Newtonian liquid. Flow becomes significant for higher values of $\beta$ rather flow for minimize values of $\beta.$ Thickness for momentum boundary layers is declined when $\beta$ is gradually enhanced. It is studied that current analysis is transformed into Newtonian model for $\beta \to \mathrm{\infty }$ while other values of $\beta$ is case of non-Newtonian model. In momentum equation, $\beta$ occurs in the denominator. Hence, an inclination in $\beta$ results velocity field is enhanced. Role ${\beta }_{e}$ (Hall parameter) is predicted on flow behavior in term of y- and x-directions. An enhancement is studied into flow (x-and y-direction) versus variation in ${\beta }_{e}.$ It is dimensionless parameter which is formulated using the theory of generalized Ohm's law in current analysis. Further, flow for platelet nanoparticles is produced significantly rather than flow for cylindrical particles. visualize the role of ion slip number $\left({\beta }_{i}\right)$ on velocity fields in terms of y- and x-directions. Similar impact is conducted when ${\beta }_{i}$ is increased. It is observed that ethylene glycol was predicted as plasma and its behavior of flow is exposed as a non-uniform magnetic field. An interaction among plasma and magnetic field produces ion slip and Hall currents which is responsible for ion slip and Hall forces. Applied magnetic field is opposite against action of ion slip and Hall forces. Hence, Lorentz force reduces when ${\beta }_{e}$ and ${\beta }_{i}$ are increased. Physically, magnetic field was placed perpendicular along the z-direction of the surface and against flow. Therefore, flow was reduced by applying higher values of Lorentz force because Lorentz force is addressed as opposite against applied magnetic field. Thickness based on flow curves for the case of platelet nanoparticles is higher than thickness generated by cylindrical nanoparticles. Role of $Fr$ on flow in y- and x-directions is addressed by . It was found that velocity distribution in y- and x-directions is decreased in presence of higher numerical values of $Fr.$ It is noticed that $Fr$ is dimensionless parameter which is produced considering Forchheirmer-porous theory. Mathematically, $Fr$ is inversely proportional versus velocity field. Consequently, an inclination in $Fr$ results flow increases in y- and x-directions. Thickness for the momentum layers is declined when $Fr$ is changed. Flow for case of $Fr = 0$ is greater than flow for case $Fr\ne 0.$

– reveal impact of temperature profile carrying three kind nanoparticles having cylindrical and platelet shapes against change in ${N}_{r}, Fr, {\beta }_{e}$ and ${\beta }_{i}$ . Reduction into temperature versus implication of ${N}_{r}$ has been estimated by . Highest performance of thermal process for case of platelet nanoparticles has been obtained rather than for cylindrical particles. It is addressed that ${N}_{r}$ is termed as dimensionless which is produced using the concept of non-uniform thermal radiation in energy equation. It was noticed that thermally energy reduces when ${N}_{r}$ is distributed. Physically, heat energy moves away from $z = 0$ along with thermal radiations. Therefore, temperature decreases at walls of plate. Thickness of thermally layers for case of platelet nanoparticles is greater than thermally thickness for case of cylindrical nanoparticles. Observations of $Fr$ on thermally field was investigated by including ternary hybrid nanoparticles. Maximum production of thermal field was estimated versus distribution in $Fr.$ Thermal enhancement for the case $Fr = 0$ is less than thermal enhancement for case of $Fr\ne 0$ . Thermal production for cylindrical particles in presence of variable thermal conductivity is less than thermal production for platelet particles. and are plotted to estimate distribution into thermal energy using parameters regarding ion slip $\left({\beta }_{i}\right)$ and Hall parameters ( ${\beta }_{e}$ ). Reduction into thermal distribution was visualized using higher values of ion slip $\left({\beta }_{i}\right)$ and Hall parameters ( ${\beta }_{e}$ ). Ion slip ( ${\beta }_{i}$ ) and Hall ( ${\beta }_{e}$ ) are opposite against Lorentz force. Further, collision rate into nanoparticles is inversely directly proportional to ion slip and Hall forces result in reduction into magnetic intensity. This reduction in magnetic intensity produces declination on thermal profile. Platelet nanoparticles experience significant thermal energy rather than experience of thermal energy for cylindrical nanoparticles. Physically, ${\beta }_{i}$ and ${\beta }_{e}$ are associated in the denominator of the Joule heating term in the energy equation. Consequent, heat energy generates because of the Joule heating process. Figure 12 reveals comparison analysis among alumina-silica-titania nanoparticles/EG, silica-titania nanoparticles/EG, silica-titania nanoparticles/EG, titania nanoparticles/EG and working fluid. It was observed that alumina-silica-titania nanoparticles/EG experiences higher thermal energy rather than for the cases of silica-titania nanoparticles/EG, silica-titania nanoparticles/EG, titania nanoparticles/EG and working fluid. Therefore, ternary hybrid nanofluid is observed as most significant for producing maximum production of thermal energy as well as reduction in thermal energy.

is prepared to determine production of wall shear stresses and thermal rate using change in Hall parameter, magnetic parameter, ion slip parameter, Casson parameter and $Fr$ . It was numerically investigated that production of thermal rate ingresses when Hall parameter, magnetic parameter, ion slip parameter are changed. However, the same behavior is investigated versus $\beta$ and thermal rate decreases versus change in $Fr$ . Wall shear stresses are inversely proportional against distribution in ion slip parameter, Casson number and Hall parameter. But inverse behavior for case of wall shear stresses is estimated when Lorentz force is changed.

Table 4. Numerically behavior of wall shear stresses and Nusselt number ^[26] with distribution in

$Fr, \beta, M, {\beta }_{e}$ and

${\beta }_{i}$ .

Parameters	Distribution in parameters	${-Re}^{1/2}Cf$	${-Re}^{1/2}Cg$	${-Re}^{-1/2}Nu$
	0.0	0.2625008163	0.4236006667	1.652101988
$Fr$	0.5	0.2701304207	0.4424323033	1.634134933
	1.4	0.2905110133	0.4903136669	1.610193943
	1.5	0.2192892414	0.2595109874	1.751988063
$\beta$	2.0	0.2403202308	0.2706220343	1.730933030
	3.5	0.2700293493	0.2903119971	1.711018177
	0.0	0.2233998059	0.2824407947	1.747439085
$M$	0.6	0.2346098044	0.2903302023	1.754403388
	1.2	0.2472144519	0.3184495583	1.780274025
	0.0	0.2437500727	0.2688997575	1.754913084
${\beta }_{e}$	0.4	0.2242905241	0.2214389298	1.764917691
	0.8	0.2146742280	0.2116991296	1.794919036
	0.0	0.2741394839	0.2805626046	1.754917615
${\beta }_{i}$	0.4	0.2341541906	0.2304551660	1.7843013073
	0.8	0.2141711628	0.2202741977	1.7899183003

| Show Table

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5. Conclusions

Thermal aspects of Casson fluid containing ternary hybrid nanofluid over stretching sheet utilizing Darcy's Forchheirmer theory are addressed. Various shapes regarding nanoparticles (platelet, cylindrical and spherical) are implemented in EG. Thermal radiation (non-linear) is utilized in the energy equation considering effects of viscous dissipation and variable wall velocity. Performance of several parameters on flow and temperature profile is addressed and numerical study is tackled by FEM. Key findings of the problem are listed below.

Hall and ion slip numbers play a remarkable role in controlling momentum and thermal boundary layers. However, declination into motion has been investigated versus $\beta$ and $Fr.$

Temperature decreases against variation in ${\beta }_{e}$ and ${\beta }_{i}$ but opposite trend is noticed in thermal layers against distribution in $Fr.$

Ethylene glycol is studied as partially ionized and shear thinning and its interaction with magnetic field produces ion slip and Hall currents. Due to this process, EG experiences ion slip and Hall forces which are against Lorentz force. Therefore, flow of EG was accelerated versus change in ${\beta }_{e}$ and ${\beta }_{i.}$

Convergence of current analysis is ensured using finite element methodology.

Maximum production for thermal energy can be achieved rather than nanofluid and hybrid nanofluid. This process is applicable in coolants (in automobiles), fuel dynamics, solar system, microelectronics, temperature reduction, cooling process and pharmaceutical process.

It was included that platelet nanoparticles experience more heat energy rather than thermal energy for cylindrical nanoparticles. Similar behavior was analyzed in velocity fields.

Nusselt number increases versus change in ion slip, Hall and $M$ parameters but thermal rate decreases versus distribution in $Fr.$

Acknowledgements

This research was supported by the Postdoctoral Research Fellowship Training Program from Khon Kaen University (PD2565-02-05).

Conflicts of interest

The authors declare that they have no conflicts of interest.

References

[1]	D. Llemit, J. M. Escaner Ⅳ, Value functions in a regime switching jump diffusion with delay market model, AIMS Math., 6 (2021), 11595–11609. http://www.aimspress.com/article/doi/10.3934/math.2021673
[2]	K. Ito, M. Nisio, On stationary solution of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1–75. https://doi.org/10.1215/kjm/1250524705 doi: 10.1215/kjm/1250524705
[3]	A. F. Ivanov, Y. I. Kazmerchuk, A. V. Swishchuk, Theory, stochastic stability and applications of stochastic delay differential equations: a survey of recent results, Differ. Equ. Dyn. Syst., 11 (2003), 55–115.
[4]	U. Küchler, E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput. Simul., 54 (2000), 189–205.
[5]	X. Mao, S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), 215–227. https://doi.org/10.1016/S0377-0427(02)00750-1 doi: 10.1016/S0377-0427(02)00750-1
[6]	G. Song, J. Hu, S. Gao, X. Li, The strong convergence and stability of explicit approximations for nonlinear stochastic delay differential equations, Numer. Algorithms, 89 (2022), 855–883. https://doi.org/10.1007/s11075-021-01137-2 doi: 10.1007/s11075-021-01137-2
[7]	S. You, L. Hu, J. Lu, X. Mao, Stabilization in distribution by delay feedback control for hybrid stochastic differential equations, IEEE Trans. Automat. Contr., 67 (2022), 971–977. https://doi.org/10.1109/TAC.2021.3075177 doi: 10.1109/TAC.2021.3075177
[8]	W. Zhang, M. H. Song, M. Z. Liu, Almost Sure Exponential Stability of Stochastic Differential Delay Equations, Filomat, 33 (2019), 789–814. https://doi.org/10.2298/FIL1903789Z doi: 10.2298/FIL1903789Z
[9]	S. Kanagawa, S. Ogawa, Numerical solutions of stochastic differential equations and their applications, Sugaku Expositions, 18 (2005), 75–99.
[10]	S. Kanagawa, The rate of convergence for approximate solutions of stochastic differential equation, Tokyo J. Math., 12 (1989), 33–48. https://doi.org/10.3836/tjm/1270133546 doi: 10.3836/tjm/1270133546
[11]	S. Kanagawa, On the rate of convergence for Maruyama's approximate solutions of stochastic differential equations, Yokohama Math. J., 36 (1988), 79–86. http://doi.org/10131/5546
[12]	H. Takahashi, Confidence intervals for Euler–Maruyama approximate solutions of stochastic delay equations, Theor. Appl. Mech. Japan, 63 (2015), 127–132. https://doi.org/10.11345/nctam.63.127 doi: 10.11345/nctam.63.127
[13]	M. Hashimoto, H. Takahashi, Interval estimations for Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays, J. Jpn. Soc. Civ. Eng., Ser. A2, 75 (2019), 105–111. https://doi.org/10.2208/jscejam.75.2_I_105 doi: 10.2208/jscejam.75.2_I_105
[14]	S. Kanagawa, Confidence intervals of discretized Euler–Maruyama approximate solutions of SDE's, Nonlinear Anal. Theory Methods Appl., 30 (1997), 4101–4104. https://doi.org/10.1016/S0362-546X(97)00159-4 doi: 10.1016/S0362-546X(97)00159-4
[15]	S. Kanagawa, S. Hasegawa, A confidence interval for the Euler–Maruyama approximate solution of SDE with unbounded coefficients, Theor. Appl. Mech. Japan, 62 (2014), 91–98. https://doi.org/10.11345/nctam.62.91 doi: 10.11345/nctam.62.91
[16]	E. Platen, N. Bruti-Liberati, Numerical solution of stochastic differential equations with jumps in finance, Stochastic Modelling and Applied Probability, volume 64, Berlin: Springer-Verlag, 2010. https://doi.org/10.1007/978-3-642-13694-8
[17]	H. Takahashi, K. Yoshihara, Approximation of solutions of multi-dimensional linear stochastic differential equations defined by weakly dependent random variables, AIMS Math., 2 (2017), 377–384. https://doi.org/10.3934/Math.2017.3.377 doi: 10.3934/Math.2017.3.377
[18]	Y. A. Alnafisah, A New Approach to Compare the Strong Convergence of the Milstein Scheme with the Approximate Coupling Method, Fractal Fract., 6 (2022), 339. https://doi.org/10.3390/fractalfract6060339 doi: 10.3390/fractalfract6060339
[19]	M. Milošević, The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments, J. Comput. Appl. Math., 298 (2016), 1–12. https://doi.org/10.1016/j.cam.2015.11.019 doi: 10.1016/j.cam.2015.11.019

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AIMS Mathematics

On the rate of convergence of Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays and their confidence interval estimations

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Abstract

1. Introduction

2. Modeling of developed model

3. Computational approach

4. Results and discussion consequences

5. Conclusions

Acknowledgements

Conflicts of interest

References

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Abstract

1. Introduction

2. Modeling of developed model

3. Computational approach

4. Results and discussion consequences

5. Conclusions

Acknowledgements

Conflicts of interest

References

AIMS Mathematics

On the rate of convergence of Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays and their confidence interval estimations

Related Papers:

Abstract

1. Introduction

2. Modeling of developed model

3. Computational approach

4. Results and discussion consequences

5. Conclusions

Acknowledgements

Conflicts of interest

References

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Abstract

1. Introduction

2. Modeling of developed model

3. Computational approach

4. Results and discussion consequences

5. Conclusions

Acknowledgements

Conflicts of interest

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