Finite Difference Technique Research Articles

Approximate solution of space fractional order diffusion equations by Gegenbauer collocation and compact finite difference scheme

In this paper, approximation of space fractional order diffusion equation are considered using compact finite difference technique to discretize the time derivative, which was then approximated via shifted Gegenbauer polynomials using zeros of (N - 1) degree shifted Gegenbauer polynomial as collocation points. The important feature in this approach is that it reduces the problems to algebraic linear system of equations together with the boundary conditions gives (N + 1) linear equations. Some theorems are given to establish the convergence and the stability of the proposed method. To validate the efficiency and the accuracy of the method, obtained results are compared with the existing results in the literature. The graphical representation are also displayed for various values of \beta Gegenbauer polynomials. It can be observe in the tables of the results and figures that the proposed method performs better than the existing one in the literature.

Numerical study of heat transfer between hot moving material and ambient medium using various hybrid nanofluids under MHD radiative-convection, viscous dissipation effects, and time-fractional condition

The heat transfer between ambient medium and moving material can be examined by the fluid flow past an impulsively started vertical plate (ISVP). In manufacturing processes such as wire/fiber drawing, hot rolling, continuous casting, and hot extrusion can be related to the hot moving material and heat transfer to the ambient medium. Also, a similar type of study addresses the understanding of aerospace engineering applications. This present study considers a mathematical model which describes the hybrid nanofluid past the ISVP by considering the transient term as a fractional derivative. The fractional order of flow governing equations is derived to enhance the heat transfer predictions with real-world problems by considering viscous dissipation, applied magnetic field, and radiation effects. Such a mathematical model is discretized by a finite difference technique and solved by a computational algorithm using FORTRAN. The findings of the study are illustrated using the velocity and temperature profiles. Also, the heat transfer and fluid friction against the boundary are interpreted using the Nusselt number and skin-friction coefficient, respectively. The results are examined under the variation of dimensionless parameters such as Eckert number, time-fractional order, Grashof number, the fraction of nanoparticles, suction, magnetic, and radiation parameters. It is observed that the heat transfer and fluid flows are affected by changing the time-fractional order. Also, a transition in the distribution of velocity and temperature is detected with varying time-fractional order.

Thermal characterization of heat source (sink) on hybridized (Cu–Ag/EG) nanofluid flow via solid stretchable sheet

Abstract The goal of this research is to consider the thermal impact on varied convection flow in hybrid nanofluids with heat generation over a two-dimensional heated flat around a stretchable sheet. The flow is considered steady and incompressible while the stretchable sheet is assumed an impermeable. Two distinctive nano-level particles are considered, namely copper (Cu) and silver (Ag) with ethylene glycol base fluid. The boundary layer was generated on a stretchable sheet surface by mixed convection flow in hybrid nanofluids. Ideally, the sink and source are thermal reservoirs of internal thermal capacities. This means you can extract or reject heat from them without changing their temperature. To make a study of thermodynamic systems like heat engines and refrigerator systems, the governing equations were solved numerically with Keller-box methodology depending on the implicit finite-difference technique. Research findings were worked with the parameters of mixed convection, Prandtl number, nanoparticle volume fraction, through various non-dimensional parameters, and heat generation. Especially for thermal generation enhancement, the fluidity and thermal dispersal get elevated. Even though the flowing behavior and the thermal dispersal of hybridity fluids with the combinations of Cu and Ag nanoparticles were similar, their values are distinct, which reflect in graphical displays. The hybrid nanofluidity gets improved with the volume variation of nanoparticles if the ϕ \phi value is 0.01 ≤ ϕ ≤ 0.05 0.01\le \phi \le 0.05 and if the flow profile value decreases ϕ h {\phi }_{{\rm{h}}} , where 0.01 ≤ ϕ h ≤ 0.05 0.01\le {\phi }_{{\rm{h}}}\le 0.05 as the dispersal of temperature enhances when the nanoparticle nanofluid constraint is improved.

Fitted finite difference technique for singularly perturbed reaction diffusion equations of third order with integral boundary condition

AbstractWe have considered a third order singularly perturbed reaction diffusion equations with integral boundary condition. On a Shishkin mesh, a numerical technique based on a finite difference scheme is provided. The proposed approach is almost first order convergent. In the discrete norm, an error estimate is obtained. Numerical examples are provided to validate the theoretical estimates.

A stable RBF-FD method for solving two-dimensional variable-order time fractional advection-diffusion equation

The meshless method is used in this paper to provide a numerical solution for a two-dimensional variable-order time fractional advection-diffusion equation. The radial basis function and the finite difference scheme on two-dimensional (2-D) arbitrary domains are the foundations of this accurate and reliable meshless approach. The name of this approach is radial basis function-generated finite differences (RBF-FD). The finite difference technique and the RBF-FD approximation are both employed for discrete time and space domains, respectively. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. The advantage of the RBF-FD method is that it has high-order convergence rates and its ability to manage sparse node layouts in arbitrary domains. The use of polyharmonic splines (PHS) as a RBF and the absence of the shape parameter are also a positive points for the mentioned method. The effectiveness and precision of the proposed method are examined with three distinct examples, incorporating two model examples and a real-world application of contamination transportation phenomena.

Effects of strain gradient on Moore-Gibson-Thompson generalized coupled non-fickian diffusion-thermoelasticity analysis in a Love-Bishop nanorod resonator: A size dependent meshless implementation

An effective meshless method based on the generalized finite difference (GFD) technique is developed to find the effects of strain gradient on the size-dependent Moore-Gibson-Thompson (MGT) generalized coupled non-Fickian diffusion-thermoelasticity analysis in a Love-Bishop nanorod resonator. A new MGT-based model with considering the strain gradient is proposed to obtain the shock-induced wave propagations in displacement, temperature and molar concentration fields in a Love-Bishop nanorod resonator for the first time. The transient governing equations of the size-dependent MGT generalized coupled non-Fickian diffusion-thermoelasticity are derived using the strain gradient equations of motion based on Love-Bishop theory, the size-dependent MGT-based energy balance equation and the size-dependent non-Fickian mass diffusion equation. The governing equations are transferred to Laplace domain and an effective GFD-based meshless method is employed to solve them. To obtain the temporal variation of fields’ variables, a proper Laplace inversion technique is employed in the problem. The small-scale effects in the MGT coupled non-Fickian diffusion-thermoelasticity analysis for the nano-sized Love-Bishop rods are taken into account using six small-scale parameters including three higher-order materials length parameters, the micro-length inertia, the micro-length thermal and the micro-length molar concentration parameters. The effects of small-scale parameters on wave propagations in all fields are discussed in detail.

J–V characteristics of dark current in truncated conical quantum dot infrared photodetectors (QDIPs)

Quantum Dot Infrared Photodetector (QDIP) is one of the promising candidates for infrared photodetection due to its controllable heterojunction bandgap and sensitivity to normal incident radiation. It is expected to be superior to infrared photodetectors of mature technologies such as Mercury Cadmium Telluride (HgCdTe) or a quantum well infrared photodetector. In the presented paper, we have developed a theoretical model for the dark current in truncated conical QDIP as the truncated conical shaped QD structure is more appropriate to describe the fabricated dots. The dark current model is based on the drift diffusion model solving the main governing Poisson’s and continuity equations. In this model, the carrier mobility is calculated by solving time-dependent Boltzmann transport equation in the photodetector material with embedded truncated conical QDs using finite difference technique. The results of the developed model have been compared with the dark current characteristics with published experimental results of Indium Arsenide/Gallium Arsenide (InAs/GaAs) truncated QDIP. The effects of QD volume, QD aspect ratio and QD density and the operating temperature on the dark current characteristics have also been investigated.

A multiconsistent computational methodology to resolve a diffusive epidemiological system with effects of migration, vaccination and quarantine

BackgroundWe provide a compartmental model for the transmission of some contagious illnesses in a population. The model is based on partial differential equations, and takes into account seven sub-populations which are, concretely, susceptible, exposed, infected (asymptomatic or symptomatic), quarantined, recovered and vaccinated individuals along with migration. The goal is to propose and analyze an efficient computer method which resembles the dynamical properties of the epidemiological model. Materials and methodsA non-local approach is utilized for finding approximate solutions for the mathematical model. To that end, a non-standard finite-difference technique is introduced. The finite-difference scheme is a linearly implicit model which may be rewritten using a suitable matrix. Under suitable circumstances, the matrices representing the methodology are M-matrices. ResultsAnalytically, the local asymptotic stability of the constant solutions is investigated and the next generation matrix technique is employed to calculate the reproduction number. Computationally, the dynamical consistency of the method and the numerical efficiency are investigated rigorously. The method is thoroughly examined for its convergence, stability, and consistency. ConclusionsThe theoretical analysis of the method shows that it is able to maintain the positivity of its solutions and identify equilibria. The method’s local asymptotic stability properties are similar to those of the continuous system. The analysis concludes that the numerical model is convergent, stable and consistent, with linear order of convergence in the temporal domain and quadratic order of convergence in the spatial variables. A computer implementation is used to confirm the mathematical properties, and it confirms the ability in our scheme to preserve positivity, and identify equilibrium solutions and their local asymptotic stability.

A Novel Fractional-Order RothC Model

A new fractional q-order variation of the RothC model for the dynamics of soil organic carbon is introduced. A computational method based on the discretization of the analytic solution along with the finite-difference technique are suggested and the stability results for the latter are given. The accuracy of the scheme, in terms of the temporal step size h, is confirmed through numerical testing of a constructed analytic solution. The effectiveness of the proposed discrete method is compared with that of the classical discrete RothC model. Results from real-world experiments show that, by adjusting the fractional order q and the multiplier term ζ(t,q), a better match between simulated and actual data can be achieved compared to the traditional integer-order model.

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.

A refined model for the effective tensile stiffness of Carbon NanoTube fibers

The paper presents a solid-mechanics based method for the determination of the macroscopic tensile properties of fibers composed of monodispersed Carbon NanoTubes (CNTs), whose length is much lower than the fiber length, arranged in a cross-sectional square lattice. In the longitudinal direction, each CNT is offset, with respect to the neighboring ones, of a given quantity. The interaction between the CNTs caps is negligible, while the model takes into account the coupling occurring on their lateral surfaces, thanks to van der Waals forces and cross-links, modeled as distributed springs. One of the main improvements with respect to previous studies is that the CNTs are here modeled as deformable elastic bars, with given axial stiffness.Under the assumption that, due to the periodicity of the CNTs arrangement, each CNT is subjected to the same loading state, it is demonstrated that the axial strain/stress fields are governed by a delayed-advanced differential equation, that is here solved by means of finite difference technique. This allows to evaluate the total axial force on the fiber, and, consequently, its effective tensile stiffness, strongly dependent on the length of the constituent CNTs, their offset and their axial compliance. Comparisons with literature data confirm the accuracy of the proposed approach.

Evanescent-field Mach-Zehnder interferometer

We propose an evanescent-field interferometer in the Mach-Zehnder configuration, which can provide an interferometric measure of the phase of evanescent fields and can be used to quantify the Hartman effect. The phase difference for such tunneled fields saturates to $\ensuremath{\pi}/2$, thus confirming the classic Hartman effect. An analytical model of the proposed evanescent-field interferometer is presented and validated by numerical simulations based on the finite-difference time-domain technique. In the numerical treatment, the requisite elements of the interferometer, such as beam splitters and mirrors, are realized through carefully designed cavities in a two-dimensional photonic crystal system. The underlying cavity modes are further engineered using a weak perturbation to spatially maneuver the evanescent field along preferred directions. The evanescent-field interferometer offers multiple avenues of introducing phase delay and has higher sensitivity that is spatially delocalized in comparison to the conventional propagating-field interferometers.

Numerical Analysis of an SEIR Epidemic Model

This paper is concerned for the numerical methods of susceptible exposed infectious recovered (SEIR) epidemic model of Coronavirus disease factor. The model is handled numerically with three extraordinary numerical scheme, forward Euler, Runge-Kutta Plan (RK-4) and the proposed non-standard finite difference (NSFD) techniques. In epidemic model of infectious illnesses, energy is a demon omen property of the consistent framework since negative worth of a subpopulation is useless. The NSFD technique ends up being more important and trustable numerical system than forward Euler and RK-4 strategies. NSFD technique uses terrifically significant properties of tireless SEIR Coronavirus scourge model like energy and presence of equilibria while forward Euler and RK-4 systems can't hold these characteristics. Furthermore, the proposed NSFD is liberated from time step size while forward Euler and RK-4 depend upon the time step size. The numerical diversions with the aid of a numerical test is presented for the endorsements of the huge number of characteristics.

Computational analysis of MHD Cross nanofluid containing gyrotactic microorganisms over a permeable horizontal cylinder through a porous medium in presence of thermal radiation and chemical reaction

In the current work, mathematical analysis of bioconvective heat and mass transfer in generalized fluid containing nanoparticles and motile gyrostatic microorganisms over a permeable horizontal cylinder embedded in a porous medium is investigated in the existence of chemical reaction, thermophoresis, and Brownian motion. Moreover, the influence of thermal radiation and magnetic field is considered. The temperature, concentration of nanoparticles, and density of microorganisms at the cylinder surface are considered nonlinear variations. The governing equations are transformed into nondimensional forms containing a set of bio-convection parameters. The transformed equations are solved numerically by a computational method composed of the finite difference technique and Newton’s linearization scheme. The computational results are validated by comparisons of some special cases of the present work with previously reported results in the literature, and an excellent agreement is obtained. The impact of included parameters is discussed and performed in graphs and tabular forms. The present study revealed that the increment in the Brownian motion leads to accelerating the velocity but reducing the temperature and the concentrations of nanoparticles and microorganisms. On the contrary, the thermophoresis force and magnetic field have the opposite effect of Brownian motion. Moreover, the rise in thermal radiation increases in velocity and temperature distributions but the opposite result occurs for nanoparticles and microorganisms concentrations. It is noticed that many important phenomena can be determined by the parameters included in the current study; therefore it has significant applications in various fields such as polymerization, petroleum industries, biotechnology, and biomedicine as in medical imaging and medicinal therapy.

Finite difference analysis of power-law nanofluid with thermal radiation and bioconvection

In this paper, we study bioconvection radiative flow of power-law nanofluid. Nanofluid characteristics are incorporated by utilizing the Buongiorno model. The influence of motile gyrotactic microorganism is retained. Magnetohydrodynamics and time-dependent aspects are under consideration. Related governing expressions are transformed into dimensionless partial differential system. The resulting nonlinear system is tackled through finite difference technique. Outcomes of emerging variables like Prandtl number, Hartmann number, Buoyancy ratio parameter, Grashof number, Peclet number, bioconvection Rayleigh number, Peclet number, radiation and Schmidt number are graphically discussed. Velocity is controlled by higher Hartman number while it boosts against higher Grashof and bioconvection Rayleigh numbers, buoyancy ratio and stretching parameters. Higher radiation, Brownian and thermophoresis parameters lead to intensification in temperature profile while it reduces for higher Prandtl number. Bioconvection profile reduces against higher Peclet and Lewis numbers.

Magnetoconvection driven melting of nanoenriched paraffin wax in cubical enclosure

The latent heat thermal energy storage using phase change material plays a vital role in thermal power production, solar energy storage and nuclear reactors. The dispersion of metallic nanoparticles into phase change material in the presence of magnetic field controls the natural convection induced melting process. This paper numerically investigates the thermo-magnetic turbulent convection induced melting of nanoenriched paraffin wax dispersed with hybrid magnetic nanoparticle pairs in a cubical enclosure with hot bottom wall. The magneto free convection effect on the melting process is investigated for high value of Hartmann number and Rayleigh number. The melting characteristics is modeled using enthalpy-porosity formulation discretized using finite difference technique and the computations are performed using an in-house three-dimensional code. It is found that the natural convection induced melting generates quadruplet thermal plume patterns from the melt zone. The dispersion of hybrid nanoparticle pairs of ferrous ferric oxide and multi-wall carbon nanotubes for particle volume fraction of 4% enhanced the melting fraction and average Nusselt number of paraffin wax by 75.21% and 42.16% respectively. The high Lorentz force developed for turbulent Hartmann number of 2000 diminished the convection induced melting phenomena. The melting time and melting rate of nanoenriched paraffin wax are a decreasing function of Hartmann number and it decreased the melt pool velocity and kinetic energy by 73.5% and 71.42%. The increase in Hartmann number decreased the average Nusselt number of nanoenhanced paraffin wax by 20.88%.

Mathematical modeling of fluid flow and pollutant transport in a homogeneous porous medium in the presence of plate stacks

This study aims at investigating the numerical analysis of pollutant transport in homogeneous porous media with solid plate stacks. The investigation was performed for solid/impervious objects of the same size placed in homogeneous porous media. The pollutant transport equation (i.e., steady-state and time dependent advection-dispersion) chosen in mathematical modeling. Three cases arise on the basis of dispersion coefficients: (a) when dispersion is uniformly constant, (b) when dispersion depends upon magnitude of the velocity, and (c) when dispersion depends upon magnitude of the velocity and directional dispersivities, all these are discussed in detail. Generally, analytical solution of such problems doesn't exist, so all the work is done numerically. The governing partial differential equation of pollutant concentration is approximated by using finite difference technique. Central, one-sided, backward and forward finite difference formulae of the same order are used to discretize the domain.Simulations of velocity potential and stream function are approximated by Matlab software. Then equipotential lines and streamlines are visualized in the form of contours. Both, velocity potential and stream function are harmonic and satisfy Laplace's equation. Fluid flow lines and pollutant concentration are represented graphically for several parameters involved in the study.It is found that entrance/exit length, shape, hydraulic conductivity, the number and position of impervious objects affect the fluid flow and pollutant transport. However, there is no significant affect of heated objects on pollutant transport. Moreover, advection and dispersion depend upon permeability of porous media and properties of solid matrix.To authenticate the Matlab scheme of finite difference, it is verified that fluid as well as pollutant fluxes (in and out) are equal. Moreover, time-dependent problem converges to steady-state form after very long time.For monitoring or forecasting the build up of contamination concentration, the pollutant transport model is considerable. As this model is affected by different parameters which are discussed above, can helps to overcome the pollutant accumulation. The solid object is main key to lessen the contamination in the underground. If the entrance or leakage point of the domain is blocked by impermeable object or filled the vertical column with material of low hydraulic conductivity it ultimately slows down or even refrains the pollutant particles to pass through. The pollutant concentration is also minimized by injecting the bioremedial agents with the help of treatment columns.

Analysis of SIRVI model with time dependent coefficients and the effect of vaccination on the transmission rate and COVID-19 epidemic waves.

COVID-19 epidemic models with constant transmission rate cannot capture the patterns of the infection data in the presence of pharmaceutical and non-pharmaceutical interventions during a pandemic. Because of this, a new modification of SIR model that contain the vaccination compartment with time dependent coefficients and weak/loss-immunity is explored. Literature review confirms that the effect of vaccination on the time dependent transmission rate is still an open problem. This study answers this open problem. In this study, we first prove the well-posedness and investigate the model dynamics to show their continuous dependence on the model parameters. We then provide an algorithm to derive the time-dependent transmission function for the epidemiologic model and the data of the infected cases. The derived coupled nonlinear differential equations show the effect of vaccination on the transmission rate. Unlike previous studies, we first filter the published data and solve the nonlinear coupled differential equations using the finite difference technique, where the coefficient of the coupled nonlinear differential equations is a function of given data. We then show that time-dependent transmission function can be represented by linear combinations of Gaussian radial base function. We then validate the prediction of our models using numerical simulations, where we used the published data of COVID-19 confirmed cases by the Ministries of Health in Saudi Arabia and Poland. Finally, the numerical solutions of a SIRVI model with time dependent transmission rate show that the waves for currently active cases are in good agreement with the data of Saudi Arabia and Poland.

A NUMERICAL SCHEME ON S-MESH FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE SOBOLEV PROBLEMS WITH LARGE TIME DELAY

The purpose of this article is to provide a numerical method for time delay singularly perturbed Sobolev type equations. First, asymptotic estimates for the Sobolev problem solution with singular perturbation and delay parameters were obtained. This estimate showed that the solution depends on the initial data. It is constructed and examined to solve this problem using a finite difference technique on a specific piecewise uniform mesh (Shishkin mesh)whose solution converges pointwise independent of the singular perturbation parameter. A discrete norm was used to investigate the stability of difference schemes. It is showed that the completely discrete scheme converges with order $O\left( \tau ^{2}+N_{l}^{-2}\ln ^{2}N_{l}\right) $ in both space and time, independent of the perturbation parameter. Finally, with a test problem and numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.

Two-grid finite difference method for 1D fourth-order Sobolev-type equation with Burgers’ type nonlinearity

This paper proposes a time two-grid finite difference (TTGFD) technique for computing numerical solution of the one-dimensional (1D) fourth-order Sobolev-type equation with Burgers-type nonlinearity. The proposed strategy mainly contains three computational stages. First, the fully nonlinear problem is approximated over a coarse grid with grid size τC. Second, an approximate solution is obtained on a fine grid with grid size τF based on the solution of the coarse grid using the Lagrangian interpolation formula. Finally, the linear problem is solved on the fine grid. Compared with the standard finite difference (SFD) scheme, an advantage of the TTGFD method is that it can maintain optimal accuracy while reducing the computational cost. Meanwhile, based on the reduced order and the discrete energy method, the conservative invariant and uniqueness of proposed method are demonstrated. In addition, the stability and convergence with the order O(τC2+τF2+h2) are evaluated in the L∞-norm, where h is the spatial step size. Finally, numerical results show the validity of the proposed strategy and support the theoretical results.