Explicit Finite Difference Technique Research Articles

Solving Time-fractional Order Radon Diffusion Equation in Water by Finite Difference Method

Objective: The aim of this research is to gain a comprehensive understanding of radon diffusion equation in water. Methods: A time fractional radon diffusion equation with Caputo sense is employed to find diffusion dynamics of radon in water medium. The fractional order explicit finite difference technique is used to find its numerical solution. A Python software is used to find numerical solution. Findings: The effect of fractional-order parameters on the distribution and concentration profiles of radon in water has been investigated. Furthermore, we study stability and convergence of the explicit finite difference method. Novelty: The fractional order explicit finite difference method can be used to estimate approximate solution of such fractional order differential equations. Keywords: Radon Diffusion Equation, Finite Difference Method, Caputo, Fractional Derivative, Python

Brownian motion and thermophoresis effects on radiative polar fluid flow with higher-order chemical reaction along a vertical porous plate

The unsteady simulation of hydromagnetic radiative polar fluid, including thermophoresis, Brownian motion, and higher-order chemical reactions across a perpendicular permeable plate with heat radiation, is performed numerically. The nonlinear PDEs are solved by the explicit finite difference technique and studio developer FORTRAN. The stability and convergence of the present model and a comparison are done to obtain accuracy. The effects of various parameters such as Brownian motion, thermophoresis, radiative heat transfer, and higher-order chemical reactions on the velocity, temperature, and concentration fields are investigated. It is found that the parameters have a significant influence on velocity and temperature profiles, angular velocity, concentration, shear stress, angular momentum density, and Nusselt and Sherwood numbers. In addition, the local share stress rises with the chemical reaction, Eckert number, and thermophoresis, whereas it decreases with the magnetic field and porous medium. In addition, the Nu (local) intensifies with higher order chemical reactions and porous mediums, whereas it falls with Eckert number, Brownian effects, micro-rotation, and thermophoresis. The concentration profile decreases with higher radiation, and the Lewis number whereas it augments with higher radiation. With an Eckert number and thermophoresis, the temperature and velocity increase. Additionally, the higher order of chemical reaction and Brownian motion augment the Sherwood number, whereas the chemical reaction, magnetic field, and porous medium reduce it. Moreover, the study may help to understand the dynamics of flow and heat transfer for the design and optimization of various engineering applications such as heat exchangers, pollutant dispersion models, energy systems, biomedical, and so on.

Numerical Simulation and Analysis of the Modified Burgers' Equation in Dusty Plasmas

This paper presents a comprehensive study of the numerical simulation of the one-dimensional modified Burgers' equation in dusty plasmas. The reductive perturbation method is employed to derive the equation, and a numerical solution is obtained using the explicit finite difference technique. The obtained results are extensively compared with analytical solutions, demonstrating a high level of agreement, particularly for lower values of the dissipation coefficient. The accuracy and efficiency of the technique are evaluated based on the absolute error. Additionally, the accuracy and effectiveness of the technique are assessed by plotting L2 and L∞ error graphs. The technique's reliability is further confirmed through von Neumann stability analysis, which indicates that the technique is conditionally stable. Overall, the study concludes that the proposed technique is successful and dependable for numerically simulating the modified Burgers' equation in dusty plasmas.

Simulation of 13 points Laplacian operator in cylindrical mesh system by using explicit finite difference technique

In nearest period of time a considerable and remarkable work have been done on the graphic design artificial intelligence, optimization, solving numerical approximation of ordinary as well as Partial differential equation by some differential operator, Discrete Laplacian operator is one of the most modern technique used for discretizatise the Ordinary differential equations (ODES) and Partial Differential equation (PDES). It is most simple technique, the researcher have taken great interest, as they may be able to find the novelty in their results specially in simulation of di-block co-polymer, solving the partial differential equation and also in finding light equation, wave equation in one as well as two dimension and etc. Discrete laplacian operator plays an essential role in the field of science and technology, shape interpolation, smoothing of the surface area, editing the meshes are the best usages of discrete laplacian operator In this study, a finite difference scheme is applied to develop the Laplacian operator using a cylindrical mesh system containing thirteen-point stencils. Laplacian operator has a significant role in the study of numerous mathematical and scientific phenomena. This employed in numerous significant computational and mathematical models, such as the Cell Dynamic Simulation system for the study of cutting-edge materials and image processing. The laplacian operator is necessary for being an isotropic discrete differential operator in a variety of models. When computing discrete isotropic results on curved surfaces, the curvilinear coordinate system may be a good option.

Improved Finite Difference Technique via Adomian Polynomial to Solve the Coupled Drinfeld’s–Sokolov–Wilson System

This study presents a new algorithm for effectively solving the nonlinear coupled Drinfeld’s–Sokolov–Wilson (DSW) system using a hybrid explicit finite difference technique with the Adomian polynomial (EFD‐AP). The suggested approach addresses the problem of accurately solving the DSW system. Numerical results are obtained by comparing the exact solution with absolute and mean square errors using a test problem to assess the EFD‐AP accuracy against the exact solution and the conventional explicit finite difference (EFD) method. The results exhibit excellent agreement between the approximate and exact solutions at different time values, and the results showed that the proposed EFD‐AP method achieves superior accuracy and efficiency compared to the EFD method, which makes it a promising method for solving nonlinear partial differential systems of higher order.

Unsteady radiative–convective flow of a compressible fluid: a numerical approach

This article is designed to inspect the thermal effects of an unsteady compressible flow of a viscous fluid through a symmetric channel. Combined effects of convective heat transfer, magnetic field, and radiation are also given special attention in this article. Basic laws of mass, momentum, and energy for compressible flow are employed in the modeling of the current problem. In addition, slip boundary conditions are also implemented in the analysis of the above thermal flow problem. Coupled nonlinear differential equations are solved numerically using explicit finite difference technique. Finally, the influence of different sundry parameters on the axial velocity, flow rate, and heat transfer are visualized through graphs. Time variant behavior of flow rate is calculated. Outcomes of the results reveal that the increment of the flow rate is related to the increase of compressibility parameters. Enhancement in the temperature profiles in the presence of radiation number is also reported. This model is the most general version of peristalsis of compressible flow in view of natural convection and radiation impact with extensive applications in aircraft industry, geophysics, and other industrial situations (cooling of electronic equipment, heat exchangers, and so forth).

Numerical exploration of MHD unsteady flow of THNF passing through a moving cylinder with Soret and Dufour effects

This experiment has studied the exploration of tangent hyperbolic nanofluid (THNF) unsteady magneto-hydrodynamic (MHD) flow with Dufour and Soret effects. The governing partial differential equations (PDE) in terms of continuity, temperature, concentration, and momentum are reformed into ordinary differential equation (ODE) form and then explained numerically by employing the explicit finite difference (EFDM) technique using Fortran programming with the help of Compaq Visual Fortran 6.6a. The behavior of the engaged physical constraints (Weissenberg number, chemical reaction parameter, Lewis number, Eckert number, Prandtl number, thermophoresis number, and Brownian parameter) on temperature, velocity, and concentration are deciphered meticulously. The outcomes indicate that adding the Soret number into the THNF generates an increment in the concentration outline. The same incident happens to Dufour’s number and temperature profile. Assessment of the current findings with earlier distributed literature is identified and observed in a good contract. Consequences of distinctive physical factors on dimensionless velocity, temperature, and concentration are particularized through graphs and tables.

NUMERICAL MODELLING OF A NON-LINEAR RADIATIVE NON-NEWTONIAN NANOFLUID FLOW WITH ARRHENIUS ACTIVATION ENERGY

This research has examined the flow properties of non-linear radiative nano non- Newtonian fluid flow via a stretched sheet, as well as the impact of Arrhenius activation energy is also inspected. The basic equations, which comprised time-dependent pivotal equations, were built using boundary layer approximations. As a numerical methodology, an explicit finite difference (EFD) technique was exerted. The fluid flow has been simulated employing FORTRAN in this case. A stability and convergence study has been performed to determine the accuracy of the numerical approach, and the system was determined to be converged at Pr≥ 0.062, and Le≥ 0.016. Here, non-dimensional outcomes based on various physical characteristics are considered. The effect of these numerous physical characteristics is explained and visually represented for a variety of flow fields. Furthermore, the examination of streamlines and isothermal lines has demonstrated enhanced visualization of the fluid flow. It has been revealed that non-linear pattern thermal radiation has a substantial impact on the heat transfer properties of nanofluid. Moreover, when non-linear radiation is addressed, the Lorentz force also has a significant impact on fluid flow. A great agreement has also served as a confirmation of the current effort. This sort of fluid has potential uses in mining, lubrication, and biomedical flow.

Simulation of magneto-hydrodynamics effects on cross fluid (blood) model with entropy generations in multiple stenosed (aneurysm) curved channel

In this letter, the non-Newtonian behavior of the blood flow is captured by using Cross fluid model within curved multiple stenosed channel. The generations of entropy within blood flow due to dissipation and magneto-hydrodynamics effects are also explored in this study. The 2-dimensional curvilinear co-ordinate system is incorporated mathematically in the fundamental proposed physical system. After then, the obtained system of equations is reduced to normalized form with the insertion of mild stenotic condition as well. Explicit finite difference technique is induced on these physical equations to acquire the numerical solution. Flow characteristics such as wall shear stress and velocity are calculated at the stenotic throat of the curved vessel in order to originate the Bejan number (Be) and Entropy generation parameter (NG) numerically. In the last section, the profiles of velocity, Bejan number, temperature and entropy generations are portrayed with the variation of different emerging parameters such as curvature Rc, Wessenburg (We) number, magnetic parameter and Brickmann number(Br). From the obtained results, it is concluded that the values of Bejan number (Be) and Entropy generation parameter (NG) are highly affected with the variation of magnetic and curvature parameters. Impacts of different parameters (such as Brickmann number (Br) and curvature effects (Rc)) on blood flow pattern are visualized in streamlines plot for general reader.

Mathematical Based Implicit and Explicit Finite Difference Techniques for Solving the Ground Water Flow Equations Using Spreadsheets

Mathematical Based Implicit and Explicit Finite Difference Techniques for Solving the Ground Water Flow Equations Using Spreadsheets

Thermal analysis of unsteady hybrid nanofluid magneto-hemodynamics flow via overlapped curved stenosed channel

In this paper, we present a detailed mathematical model of the unsteady hybrid blood through the curved overlapping stenosed vessels in the presence of the applied external magnetic field. The governing differential equations of this model are derived from the generic laws of conservation of momentum and energy. These derived differential equations are then rendered dimensionless by incorporating the normalization parameters, which are subsequently followed by the implementation of mild stenotic supposition. The parabolic equations are solved numerically by using the explicit finite difference technique. After that, these numerical calculations are used to simulate several flow characteristics such as velocity, impedance, and wall shear stress using the set of emerging parameters taken from literature. The influence of these emerging parameters on the flow characteristics reveals that the velocity of the hybrid blood flow stream decelerates due to the presence of an induced magnetic field having the inclusion of nanoparticles. Finally, the global behavior of hybrid blood flow within the curved stenosed channels is sketched and analyzed by using various streamline plots.

Chemically Reactive MHD Eyring-Powell Nanofluid Flow past a Stretching Surface with Convergence Test

The motive of this work is to examine the transfer of heat as well as mass phenomena of Eyring-Powell fluid flow on a stretching type of porous medium. The impact of heat absorption and thermal radiation are also considered here to describe the characteristics of the fluid flow. Firstly, a mathematical model of this fluid flow is established which includes time dependent continuity, energy, momentum and concentration formula. Then the fundamental equations are formed to dimensionless format. After that an explicit type finite difference technique is imposed to solve the model by taking the help of FORTRAN. The stability as well as convergence analysis (SCA) is used to develop and check the precession of the overall system. Moreover, the fields of temperature, velocity, concentration, skin friction, Nusselt number and Sherwood number got affected significantly with the influences of various pertinent parameters which are presented in different color diagrams. However, the updated visualization of the fluid flows is also presented by both isotherms and streamlines for the impact of radiation parameter. Moreover, for Eyring-Powell fluid, an interesting observation has been made that the thermophoretic parameter is influencing the temperature field significantly rather than the Brownian parameter. Finally, for the validation of the ongoing investigation, a suitable comparison is also depicted with some published papers and a proper agreement is noticed.

Simulations of unsteady blood flow through curved stenosed channel with effects of entropy generations and magneto-hydrodynamics

This research study explores the effects of entropy generations on the pulsatile blood flow through a w-shape curved stenosed channel. The mathematical formulations of this physical problem are derived from the couple equations of momentum and energy. These equations are first normalized, and then solved numerically using the explicit finite difference technique. Subsequently, the solutions of these equations are utilized in the calculation of the Bejan number (Be) and entropy generations (NG). To simplify this problem, a few assumptions are taken into consideration; for instance, a mild stenotic condition is assumed in order to reduce the order of differential equations. The results are based on various velocity graphs sketched under the influence of the curvature (Rc) and magnetic (M) parameters. The magnitude of velocity as well as streamline profiles are highly affected due to the curvature effects of channel wall. The results demonstrate that the value of Bejan (Be) number is highly influenced by increasing the value of curvature parameter (Rc) as depicted in the graphs. Similarly, the shape of velocity profile reduces to symmetric in the unbent vessel. Moreover, it is also observed that the magnitude of velocity has decelerated in the presence of magnetic field.

Numerical Solution for Fuzzy Diffusion Problem via Two Parameter Alternating Group Explicit Technique

The computational technique has become a significant area of study in physics and engineering. The first method to evaluate the problems numerically was a finite difference. In 2002, a computational approach, an explicit finite difference technique, was used to overcome the fuzzy partial differential equation (FPDE) based on the Seikkala derivative. The application of the iterative technique, in particular the Two Parameter Alternating Group Explicit (TAGE) method, is employed to resolve the finite difference approximation resulting after the fuzzy heat equation is investigated in this article. This article broadens the use of the TAGE iterative technique to solve fuzzy problems due to the reliability of the approaches. The development and execution of the TAGE technique towards the full-sweep (FS) and half-sweep (HS) techniques are also presented. The idea of using the HS scheme is to reduce the computational complexity of the iterative methods by nearly/more than half. Additionally, numerical outcomes from the solution of two experimental problems are included and compared with the Alternating Group Explicit (AGE) approaches to clarify their feasibility. In conclusion, the families of the TAGE technique have been used to overcome the linear system structure through a one-dimensional fuzzy diffusion (1D-FD) discretization using a finite difference scheme. The findings suggest that the HSTAGE approach is surpassing in terms of iteration counts, time taken, and Hausdorff distance relative to the FSTAGE and AGE approaches. It demonstrates that the number of iterations for HSTAGE approach has decreased by approximately 71.60-72.95%, whereas for the execution time, the implementation of HSTAGE method is between 74.05-86.42% better. Since TAGE is ideal for concurrent processing, this method has been seen as the key benefit as it consumes sets of independent tasks that can be performed at the same time. The ability of the suggested technique is projected to be useful for the advanced exploration in solving any multi-dimensional FPDEs.

Numerical simulation of periodic MHD casson nanofluid flow through porous stretching sheet

The perspective of this paper is to characterize a Casson type of Non-Newtonian fluid flow through heat as well as mass conduction towards a stretching surface with thermophoresis and radiation absorption impacts in association with periodic hydromagnetic effect. Here heat absorption is also integrated with the heat absorbing parameter. A time dependent fundamental set of equations, i.e. momentum, energy and concentration have been established to discuss the fluid flow system. Explicit finite difference technique is occupied here by executing a procedure in Compaq Visual Fortran 6.6a to elucidate the mathematical model of liquid flow. The stability and convergence inspection has been accomplished. It has observed that the present work converged at, Pr ≥ 0.447 indicates the value of Prandtl number and Le ≥ 0.163 indicates the value of Lewis number. Impact of useful physical parameters has been illustrated graphically on various flow fields. It has inspected that the periodic magnetic field has helped to increase the interaction of the nanoparticles in the velocity field significantly. The field has been depicted in a vibrating form which is also done newly in this work. Subsequently, the Lorentz force has also represented a great impact in the updated visualization (streamlines and isotherms) of the flow field. The respective fields appeared with more wave for the larger values of magnetic parameter. These results help to visualize a theoretical idea of the effect of modern electromagnetic induction use in industry instead of traditional energy sources. Moreover, it has a great application in lung and prostate cancer therapy.

Soret and Dufour Effects on Unsteady Free Convection Fluid Flow in the Presence of Hall Current and Heat Flux

Unsteady MHD natural convective heat and mass transfer flow through a semi-infinite vertical porous plate in a rotating system have been investigated with the combined Soret and Dufour effects in the presence of Hall current and constant heat flux. It is considered that the porous plate is subjected to constant heat flux. The obtained non-dimensional, non-similar coupled non-linear and partial differential equations have been solved by explicit finite difference technique. Numerical solutions for velocities, temperature and concentration distributions are obtained for various parameters by the above mentioned technique. The local and average shear stresses, Nusselt number as well as Sherwood number are also investigated. The stability conditions and convergence criteria of the explicit finite difference scheme are established for finding the restriction of the values of various parameters to get more accuracy. The obtained results are illustrated with the help of graphs to observe the effects of various legitimate parameters.

Mathematical Modelling and Numerical Simulation of Mass Transfer During Deep-Fat Frying of Plantain (Musa paradisiacal AAB) Chips (ipekere)

Abstract This study developed a mathematical model following the fundamental principles of mass transfer for the simulation of the oil and moisture content change during the Deep-Fat Frying of plantain (ipekere) chip. The explicit Finite Difference Technique (FDT) was used to conduct a numerical solution to the consequential governing equation (partial differential equation) that was used to describe the mass transfer rate during the process. Computer codes that were computed in MATLAB were used for the implementation of FDT at diverse frying conditions. Samples of the plantain were cut into portions of 2 mm thickness, and these sliced portions were fried at separate frying oil temperatures (170, 180 and 190°C) between 0.5 and 4 minutes. The experimental data and the predicted outcomes were compared for the validation of the model, and the juxtaposition revealed a plausible agreement. The predicted values and the experimental values of oil and moisture transfer models produced correlation coefficients that range from 0.96 to 0.99 and 0.94 to 0.99, respectively. The predicted outcomes could be utilized for the control and design of the DFF.

A Mathematical Model of Horizontal Averaged Groundwater Pollution Measurement with Several Substances due to Chemical Reaction

Chloride is a well-known chemical compound that is very useful in industry and agricultural, chloride can be transformed to hypochlorite, chlorite, chlorate and perchlorate, chloride and their substances are not dangerous if we used in the optimal level. Groundwater that contaminated chloride and their substances impacts human health, for an example, if we drink water that contaminated chloride exceed 250 mg/L it can cause heart problems and contribute to high blood pressure. to avoid this problem, we used mathematical models to explain groundwater contamination with chloride and their substances. Transient groundwater flow model provides the hydraulic head of groundwater, in this model we will get the level of groundwater, next, we need to find its velocity and direction by using the result in first model put into second model. Groundwater velocity model provides x- and z-direction vector in groundwater, after computation we will plugin the result into the last model to approximated the chloride concentration in groundwater. Groundwater contamination dispersion model provides chloride, hypochlorite, chlorite, chlorate and perchlorate concentration. The proposed explicit finite difference techniques are used to approximate the model solution. Explicit method was used to solved hydraulic head model. Forward space described groundwater velocity model. Forward time and central space used to predict transient groundwater contaminated models. The simulations can be used to indicate when each simulated zone becomes a hazardous zone or a protection zone.

Numerical Simulation of a Two-Dimensional Vertically Averaged Groundwater Quality Assessment in Homogeneous Aquifer Using Explicit Finite Difference Techniques

Leachate contamination in a landfill causes pollution that flowing down to the groundwater. There are many methods to measure the groundwater quality. Mathematical models are often used to describe the groundwater flow. In this research, the affection of landfill construction to groundwater-quality around rural area is focused. Three mathematical models are combined. The first model is a two-dimensional groundwater flow model. It provides the hydraulic head of the groundwater. The second model is the velocity potential model. It provides the groundwater flow velocity. The third model is a two-dimensional vertically averaged groundwater pollution dispersion model. The groundwater pollutant concentration is provided. The forward time centered technique with the centered in space, the forward in space and the backward in space with all boundaries are used to obtain approximate hydraulic head, the flow velocity in x- and y- directions, respectively. The approximated groundwater flow velocity is used to input into a two-dimensional vertically averaged groundwater pollution dispersion model. The forward time centered space technique with the centered in space, the forward in space and the backward in space with all boundaries are used to obtain approximate the groundwater pollutant concentration. The proposed explicit forward time centered spaced finite difference techniques to the groundwater flow model the velocity potential model and the groundwater pollution dispersion model give good agreement approximated solutions.

A Study of Two Dimensional Unsteady MHD Free Convection Flow over a Vertical Plate

In this paper, unsteady free convection heat transfer flow over a vertical plate in the presence of a magnetic field is discussed in detail. The dimensionless partial differential equations of continuity, momentum along energy are analyzed with suitable transformations. For numerical calculation, an implicit finite difference method is applied to solve a set of nonlinear dimensionless partial differential equations. Dimensionless velocity and temperature profile are also investigated due to the effects of assumed parameters in the concerned problem. An explicit finite difference technique is used to compute velocity and temperature profiles. The stability conditions are also examined.