Adaptive step size stochastic runge-kutta method of order 1.5(1.0) for stochastic differential equations (SDEs)

The stiff stochastic differential equations (SDEs) involve the solution with sharp turning points that permit us to use a very small step size to comprehend its behavior. Since the step size must be set up to be as small as possible, the implementation of the fixed step size method will result in high computational cost. Therefore, the application of variable step size method is needed where in the implementation of variable step size methods, the step size used can be considered more flexible. This paper devotes to the development of an embedded stochastic Runge-Kutta (SRK) pair method for SDEs. The proposed method is an adaptive step size SRK method. The method is constructed by embedding a SRK method of 1.0 order into a SRK method of 1.5 order of convergence. The technique of embedding is applicable for adaptive step size implementation, henceforth an estimate error at each step can be obtained. Numerical experiments are performed to demonstrate the efficiency of the method. The results show that the solution for adaptive step size SRK method of order 1.5(1.0) gives the smallest global error compared to the global error for fix step size SRK4, Euler and Milstein methods. Hence, this method is reliable in approximating the solution of SDEs

Performance of 5-stage, 4-stage and specific stochastic Runge-Kutta methods in approximating the solution of stochastic biological model

In recent years, the transition on modelling physical systems via stochastic differential equations (SDEs) has attracted great interest among researchers. This is due to the limitations of ordinary differential equations in presenting the real phenomenon. To the fact that the stochastic models incorporate the random effects that may influence the behaviour of physical systems, SDEs seems to be the best model that can be used i n assessing those systems. The growing interest among researchers in modelling the systems via SDEs comes with the rise in the need of numerical methods to approximate the solutions for SDEs. This is because by taking into account the random fluctuations in SDEs resulting to the complexity of finding the exact solution of SDEs. Therefore, it contribute to the increasing number of research to decide on the best numerical approach to solve the systems of SDEs. This paper is devoted to investigate the performance of 5-stage stochastic Runge-Kutta ( SRK5) with order 2.0, 4-stage stochastic Runge-Kutta ( SRK4), specific stochastic Runge-Kutta with order 1.5 ( SRKS1.5) and commutative specific stochastic Runge-Kutta with order 1.5 (SRKST2) in approximating the solution of stochastic model in biological system. A comparative study of SRK5, SRK4, SRKS1.5 and SRKST2 methods will be presented in this paper. The linear SDE model and the stochastic model of C. Acetobutylicum cell growth will be used to examine the performance of those methods and the numerical experiment will be conducted. The numerical solutions obtained will be discussed

Eyring-powell hybrid nanofluid with radiative heat flux: Case over a shrinking sheet

This study investigates the radiative heat flux effect on Eyring-Powell hybrid nanofluid together with analysis thermal passing towards a shrinking sheet containing nanoparticles. The proposed mathematical model respected to the boundary condition is converted to the similarity equations by adopting the suitable transformations. In order to reduce the complexity of model, the similarity equations are then computed by applying the bvp4c function in MATLAB software. Outcomes reveal the thermal performance is upgraded in dispersing the nanoparticle to the base fluid, and it is marked under hybrid nanoparticles consideration. The presence of Eyring-Powell, radiation and shrinking parameters are the controller parameter in regulatory the sheer stress and thermal performance of the fluid. Ag − MgO

Mathematical Model of Simultaneous Flow between Casson Fluid and Dust Particle over a Vertical Stretching Sheet

The process involving multiphase flow generally can be found in natural phenomena and many industrial applications. It might be between solid-liquid flow, liquid-liquid flow, gas-solid flow or gas-liquid flow. Their interaction is significant and able to influence the flow characteristics. The experimental work for this topic has been widely performed in order to obtain the best interaction output but it is still incomplete due to the limitation in term of cost and safety issue. Since then, the mathematical model is proposed to counter that constraint. This paper is aims to propose the mathematical model representing the two-phase flow which the interaction of non-Newtonian Casson fluid and solid particles is examined. The flow is investigated moving over a stretching sheet and the combined convection is considered together with the influence of heat generation in fluid phase. The governing equations representing the two-phase model are first transformed into the ordinary differential equations using established similarity transformations where the complexity of the model is reduced. The resulting equations are then solved by employing the Keller-box method with the help of Matlab software. The numerical output in term of velocity and temperature distribution for both phases are illustrated graphically and the value of skin friction and heat transfer coefficients are presented in tabular form for various value of mixed convection parameter, heat generation parameter and Casson parameter. Findings reveal that, the parameter under investigation affects the flow characteristics and present a significant impact to both phases except for mixed convection parameter

Analysis of convective transport of temperature-dependent viscosity for non-newtonian erying powell fluid: A numerical approach

Non-Newtonian is a type of fluid that does not comply with the viscosity under the Law of Newton and is being widely used in industrial applications. These include those related to chemical industries, cosmetics manufacturing, pharmaceutical field, food processing, as well as oil and gas activities. The inability of the conventional equations of Navier–Stokes to accurately depict rheological behavior for certain fluids led to an emergence study for non-Newtonian fluids’ models. In line with this, a mathematical model of forced convective flow on non-Newtonian Eyring Powell fluid under temperature-dependent viscosity (TDV) circumstance is formulated. The fluid model is embedded with the Newtonian heating (NH) boundary condition as a heating circumstance and is assumed to move over a stretching sheet acting vertically. Using appropriate similarity variables, the respective model was converted into ordinary differential equations (ODE), which was later solved utilizing the Keller box approach. The present model is validated by comparing the existing output in literature at certain special limiting cases, where the validation results display a firm agreement. The current outputs for the proposed model are shown in tabular and graphical form for variation of skin friction plus Nusselt number, velocity and temperature distribution, respectively

Thermally Radiation of MHD Two-Phase Williamson Fluid with Newtonian Heating (NH)

Due to the unique qualities in its behaviours that study the solid and fluid aspects, a study on twophase flow (solid and liquid) is deemed to be supplementary trustworthy in describing the application of liquid in industrial sectors. Over the past few decades, several non-Newtonian fluid models have been found, but the Williamson model stands out as the most intriguing. The Williamson flow model will be more fascinating to study when the existing particles are considered. Therefore, the purpose of this article is to investigate Williamson fluid with dust particles under the existing MHD and heat radiation. To make inquiry more intriguing, the analysed model also included a Newtonian heating (NH) condition. After employing similarity transformation, the resultant equations, as in Ordinary Differential Equations (ODEs), are solved using the Runge-Kutta Fehlberg (RKF45) method. The findings showed that the presence of fluid-particle interaction (FPI) influenced the fluid velocity, subsequent in a declining the fluid movement and an increase in particle motio

How fluid particle interaction affects the flow of dusty williamson fluid

A model of two-phase flow involving non-Newtonian fluid is described to be more reliable to present the fluid that involves industrial applications due to the special characteristics in its behavior. Many models of non-Newtonian fluid were discovered in the last few decades but the model that captured the most attention is the Williamson model. The consideration of the existing particles in the Williamson flow (two-phase Williamson fluid) will make the model more interesting to investigate. Hence, this paper is aimed to explore the flow of two-phase Williamson fluid model in the presence of MHD and thermal radiation circumstances. The obtained ordinary differential equations after the transformations are solved using the Runge-Kutta Fehlberg (RKF45) method. The flow is considered asymmetric since it moves over a vertical stretching sheet with external stimuli. The result displays variation in dust phases compared to the fluid phase under distribution of velocity and temperature. It can be concluded that the fluid–particle interaction (FPI) parameter lessening the motion of fluid and heating characteristics. In addition, the upsurges on skin friction and heat transfer are resulting from the rising FPI. Furthermore, the presence of Williamson parameter increases the skin friction while causing degenerations on heat transfer of flow

Two-phase flow of non-newtonian eyring fluid over a vertical stretched surface with temperature dependent viscosity

The investigation of the fluid flow problem via mathematical approach for non-Newtonian fluid is challenging due to the rise in complexity in its model. However, the study still attracted researchers since the model is able to capture properties of the existing fluid involved in industrial applications. There are several models representing the non-Newtonian fluid. In this paper, the model of Eyring-Powell fluid with dust particle under influence of temperature dependent viscosity is discussed. The model is formulated using the law of conservation of mass, the first law of thermodynamics and Navier-Stokes equation. The complexity of the model is reduced to a set of ordinary differential equations and the computation is done by using the finite difference method. The validation of the present results is attained by direct comparison with those existing in literature which is found to be in excellent agreement. The investigation revealed the viscosity of the fluid affecting the flow characteristics in both the phases

Numerical solutions on Reiner-Philippoff (RP) fluid model with velocity and thermal slip boundary condition

Non-Newtonian fluid model was created against the Newton’s Law of viscosity where the viscosity is no more constant and dependent on the shear rate. The existing such fluid can be found in many industrial claims especially in food manufacturing, lubrication, biomedical flows and oil and gas. Besides, the used of non Newtonian fluid occurs in mining industry where the slurries and muds are often handled. There are many models on non-Newtonian fluid available in literature where some of them capture the specific properties. The Reiner–Philippoff (RP) fluid model is considered in this endeavour due to the capabilities of the model which can be acted in three different family of fluid which are viscous, shear thickening and the shear-thinning. Mathematical model is constructed using continuity, momentum and energy equations where in form of partial differential equations (PDEs). The complexity of the proposed model is abridged by deduced the equations into ordinary differential equations (ODEs) by adopting similarity variablesbefore the computation is done by bvp4c function drive in MATLABsoftware. To ratify the validity of the proposed model as well as numerical outputs, the comparative study is performed and it found to be in very strong agreement under limiting case where the present model is condensed to be identical with the reported model previously. The consequences of pertinent parameters on fluid’s characteristics are analyzed in details through the plotted graphic visuals and tabular form

Numerical Solutions on Reiner–Philippoff (RP) Fluid Model with Velocity and Thermal Slip Boundary Condition

Non-Newtonian fluid model was created against the Newton’s Law of viscosity where the viscosity is no more constant and dependent on the shear rate. The existing such fluid can be found in many industrial claims especially in food manufacturing, lubrication, biomedical flows and oil and gas. Besides, the used of non-Newtonian fluid occurs in mining industry where the slurries and muds are often handled. There are many models on non-Newtonian fluid available in literature where some of them capture the specific properties. The Reiner–Philippoff (RP) fluid model is considered in this endeavour due to the capabilities of the model which can be acted in three different family of fluid which are viscous, shear thickening and the shear-thinning. Mathematical model is constructed using continuity, momentum and energy equations where in form of partial differential equations (PDEs). The complexity of the proposed model is abridged by deduced the equations into ordinary differential equations (ODEs) by adopting similarity variables before the computation is done by bvp4c function drive in MATLAB software. To ratify the validity of the proposed model as well as numerical outputs, the comparative study is performed and it found to be in very strong agreement under limiting case where the present model is condensed to be identical with the reported model previously. The consequences of pertinent parameters on fluid’s characteristics are analyzed in details through the plotted graphic visuals and tabular form. © 2022, Penerbit Akademia Baru. All rights reserved