Forward Difference Method Research Articles

On the exact and numerical solutions to the FitzHugh–Nagumo equation

In this paper, with the help of a computer package program, the auto-Bäcklund transformation method (aBTM) and the finite forward difference method are used for obtaining the wave solutions and the numeric and exact approximations to the FitzHugh–Nagumo (F-N) equation, respectively. We successfully obtain some wave solutions to this equation by using aBTM. We then employ the finite difference method (FDM) in approximating the exact and numerical solutions to this equation by taking one of the obtained wave solutions into consideration. We also present the comparison between exact and numeric approximations and support the comparison with a graphic plot. Moreover, the Fourier von-Neumann stability analysis is used in checking the stability of the numeric scheme. We also present the [Formula: see text] and [Formula: see text] error norms of the solutions to this equation.

A numerical efficient splitting method for the solution of two dimensional susceptible infected recovered epidemic model of whooping cough dynamics: Applications in bio-medical engineering

Background and Objective The positivity property of the non-linear dynamical systems is one of the essential features in different fields of bio-medical engineering, science and many more. The state variables, involving in the models, describing the natural phenomenon such as concentration, density and population size etc. must be positive. Therefore, the computing techniques used to solve the system of non-linear differential equations must be consisted with the continuous nature of the models. But, unfortunately there are some existing techniques in the literature that do not preserve the positivity property, especially for the multi-space dimensional models. So there is a gap in the literature that should be filled up, by constructing the positivity preserving numerical algorithms. In this study, we consider a susceptible-infected-recovered (SIR) reaction diffusion epidemic model in two space dimensions from biomedical engineering and solved numerically to observe the behavior of the model. Since the state variables involved in this system are population densities therefore we design a novel computational method which is time efficient because of its splitting structure and holds the positivity as well as other important structure of epidemic system. Methods Three different computational techniques are designed to examine the numerical solution of SIR model of infectious disease. Two approaches are well-known existing computing methods named as forward Euler finite difference (FD) method and backward Euler operator splitting finite difference (OS-FD) method. The third approach is operator splitting nonstandard finite difference (OS-NSFD) method which is devised by using the NSFD rules. Results The proposed OS-NSFD technique retains efficiently the stability of equilibria as well as the positivity. Graphical behavior depicts that the existing computing methods can not get success to preserve the structure of the epidemic system of whooping cough dynamics. At the same time OS-NSFD computing method is proven to be reliable and suitable for the system of bio-medical engineering mathematically and graphically. Conclusion A reliable and novel computing technique is developed for the solution of two dimensional reaction diffusion problem. This technique preserves all the imperative characteristics of the model under study. Also the time efficiency of this method makes it easy to find the solution of physical system in two space dimension. The comparison with other techniques shows the efficacy and reliability of the designed technique.

Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation

In this paper, present solution of one-dimensional linear parabolic differential equation by using Forward difference, backward difference, and Crank Nicholson method. First, the solution domain is discretized using the uniform mesh for step length and time step. Then applying the proposed method, we discretize the linear parabolic equation at each grid point and then rearranging the obtained discretization scheme we obtain the system of equation generated with tri-diagonal coefficient matrix. Now applying inverse matrixes method and writing MATLAB code for this inverse matrixes method we obtain the solution of one-dimensional linear parabolic differential equation. The stability of each scheme analyses by using Von-Neumann stability analysis technique. To validate the applicability of the proposed method, two model example are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (E∞) and Root mean error (E2). Also, condition number (K(A)) and Order of convergence are calculated. The stability of this Three class of numerical method is also guaranteed and also, the comparability of the stability of these three methods is presented by using the graphical and tabular form. The proposed method is validated via the same numerical test example. The present method approximate exact solution very well.

Carbon dioxide transport in radial miscible flooding in consideration of rate-controlled adsorption

As an important technology in improving oil recovery, CO2 miscible flooding has achieved great success in many oil fields. However, studies on CO2 transport in porous media are mostly based on equilibrium adsorption in one-dimensional displacement where rate-controlled adsorption is not taken into consideration. In this work, we develop a radial convection–dispersion model, which simultaneously combines rate-controlled adsorption, convection, and dispersion. Based on this model, four dimensionless groups representing CO2 dispersion, adsorption capacity, flow rate, and kinetic rate groups are proposed for the first time. Subsequently, the Barakat–Clark forward and backward difference methods are combined to solve the mathematical model. CO2 concentration and adsorption at different positions in the porous media (including the effluent concentration and adsorption histories) are then calculated. Furthermore, the effects of the parameters on CO2 transport behavior are studied in detail. The results reveal that CO2 gradually moves forward with the increase in the CO2 injection volume. Once CO2 reaches a certain position, its concentration there increases and an S-shaped curve is formed. Moreover, the adsorption capacity at this position also increases significantly, and the changing rate is much higher than CO2 concentration. CO2 effluent concentration grows more uniform, and the breakthrough occurs earlier with increased dispersion velocity and CO2 injection rate and decreased adsorption capacity and adsorption rate. The mathematical model developed in this study is of great importance in predicting CO2 transport behavior in radial miscible flooding.

A Discrete-Time State Estimation for Nonlinear Systems With Noises

This paper proposes a discrete-time observer to estimate states of continuous-time nonlinear systems with presences of noises. Most conventional discrete-time state observers are designed using discrete-time models, which are derived by applying the forward-difference method to continuous-time models without considering their poor accuracy. This poor accuracy of the discretization may affect the accuracy of the observer-based on it and lead to a misevaluation. The proposed observer is based on an application of the extended Kalman filter (EKF) for a discrete-time model derived by a discretization method called continualized discretization. The proposed observer was applied to estimate the states of the Lorenz and van der Pol oscillators, which have complex dynamics such as limit cycle and chaos. The simulation results showed that the proposed observer gives better performances in estimating the real state and retaining the system dynamics of the original continuous-time model than the conventional method.

Numerical analysis of diffusive susceptible-infected-recovered epidemic model in three space dimension

Abstract In this article, numerical solution of three dimensional susceptible-infected-recovered (SIR) reaction-diffusion epidemic system is furnished with a time efficient operator splitting nonstandard finite difference (OS-NSFD) method. We perform the comparison of proposed OS-NSFD method with popular forward Euler explicit finite difference (FD) method and time efficient backward Euler operator splitting finite difference (OS-FD) implicit method. The proposed OS-NSFD method is implicit in nature but computationally efficient as compared to forward Euler explicit (FD) scheme. The numerical stability and bifurcation value of transmission coefficient for SIR reaction-diffusion epidemic system is also investigated with the aid of Routh–Hurwitz method. At the end, we give two numerical experiments and simulation. In first experiment, all the numerical schemes are compared with the help of simulations. In second experiment we show the simulations of proposed NSFD technique at different values of parameters. Also we discuss the importance of transmission rate to control the spread of disease with the help of simulations.

Algebra of Fuzzy Random Variable and its Application to Science and Engineering Problem- Radon Transport from Soil into Building

The theme of this paper is to bring out the mathematical details of fuzzy random variable and its application to model radon transport from soil into buildings. The physical processes that influence radon transport are advection and diffusion. Parameters associated with the governing partial differential equation describing radon transport from soil into buildings are radon diffusion coefficient and advective velocity of radon in air. Both the parameters are imprecise. Imprecision of these parameters is addressed as fuzzy random variable due to presence of inherent randomness within their fuzziness. Handling fuzziness within randomness is an innovative concept of computation to quantify the uncertainty of any physical system. Paper presents the mathematical structure (algebra) of fuzzy random variables. The concept of fuzzy randomness is implemented in developing radon transport model. Numerical solution of radon transport model with fuzzy random parameters is obtained by explicit forward time central space finite difference method. Support, uncertainty index, possibility, necessity and credibility of the spatial and temporal behaviour of the random behaviour of radon concentration are also computed to explore the concept and role of the fuzzy randomness in decision making issue

Numerical approximation of the fractional HIV model using the meshless local Petrov\u2013Galerkin method

This paper deals with the model of fractional HIV-1 infection of CD4+T cells transformation with homogeneous Neumann boundary conditions. Numerical methods for solving fractional time differential equations are developed with Caputo’s definition. The forward difference methods were constructed applied to the approximation of the fractional time differential equation. The MLPG method is used to solve the problem of fractional HIV models for spatial discretization. Approximated solutions at the time level n use conventional iterative methods such as fixed point iterations to handle the nonlinear parts. An analysis of stability and convergence of numerical schemes is presented along with the eigenvalue of the matrix. The abilities of the developed formula was confirmed through four numerical examples base on convergence and accuracy of numerical results. The results of the numerical experiments were compared with the solution of the integer order differential equation to confirm the accuracy and efficiency of the proposed scheme. The simulation results show that the formula is easy to use and useful for those interested in fractional derivatives.

Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation

Radial basis function-based quasi-interpolation performs efficiently in high-dimensional approximation and its applications, which can attain the approximant and its derivatives directly without solving any large-scale linear system. In this paper, the bivariate multi-quadrics (MQ) quasi-interpolation is used to simulate two-dimensional (2-D) Burgers’ equation. Specifically, the spatial derivatives are approximated by using the quasi-interpolation, and the time derivatives are approximated by forward finite difference method. One advantage of the proposed scheme is its simplicity and easy implementation. More importantly, the proposed scheme opens the gate to meshless adaptive moving knots methods for the high-dimensional partial differential equations (PDEs) with shock or soliton waves. The scheme is also applicable to other non-linear high-dimensional PDEs. Two numerical examples of Burgers’ equation (shock wave equation) and one example of the Sine–Gordon equation (soliton wave equation) are presented to verify the high accuracy and efficiency of this method.

Clothing Design Model for High Temperature Work based on Heat Conduction

By analyzing the temperature changes of the three-layer high-temperature clothing and the air layer of the fourth layer, according to the calculation of the temperature distribution of the high-temperature clothing, the temperature changes of the high-temperature clothing are related to the heat, ignoring the heat transfer of heat radiation in the clothing and the internal heat sources between the layers, defining three variables of time, temperature and thickness, according to the law of conservation of energy and Fourier’s law of heat transfer. The law obtains the heat conduction equation. The forward difference method in the finite difference method is used to solve the equation, and the numerical model is obtained. Then, the relationship diagrams of temperature, time and thickness are drawn. The experimental results show that the heat absorbed by the outside is consumed layer by layer in the process of heat transfer, and finally the air layer contacted with human body tends to 48 C. Finally, according to the formula Q = cm Delta U of heat absorption in physics, further analysis is carried out, and the optimal thickness of Layer II and Layer IV is obtained by using the optimization algorithm.

Modeling of soft tissue thermal damage based on GPU acceleration

Hyperthermia treatments require precise control of thermal energy to form the coagulation zones which sufficiently cover the tumor without affecting surrounding healthy tissues. This has led modeling of soft tissue thermal damage to become important in hyperthermia treatments to completely eradicate tumors without inducing tissue damage to surrounding healthy tissues. This paper presents a methodology based on GPU acceleration for modeling and analysis of bio-heat conduction and associated thermal-induced tissue damage for prediction of soft tissue damage in thermal ablation, which is a typical hyperthermia therapy. The proposed methodology combines the Arrhenius Burn integration with Pennes’ bio-heat transfer for prediction of temperature field and thermal damage in soft tissues. The problem domain is spatially discretized on 3-D linear tetrahedral meshes by the Galerkin finite element method and temporally discretized by the explicit forward finite difference method. To address the expensive computation load involved in the finite element method, GPU acceleration is implemented using the High-Level Shader Language and achieved via a sequential execution of compute shaders in the GPU rendering pipeline. Simulations on a cube-shape specimen and comparison analysis with standalone CPU execution were conducted, demonstrating the proposed GPU-accelerated finite element method can effectively predict the temperature distribution and associated thermal damage in real time. Results show that the peak temperature is achieved at the heat source point and the variation of temperature is mainly dominated in its direct neighbourhood. It is also found that by the continuous application of point-source heat energy, the tissue at the heat source point is quickly necrotized in a matter of seconds, while the entire neighbouring tissues are fully necrotized in several minutes. Further, the proposed GPU acceleration significantly improves the computational performance for soft tissue thermal damage prediction, leading to a maximum reduction of 55.3 times in computation time comparing to standalone CPU execution.

Research on a novel fractional GM(α, n) model and its applications

PurposeThe purpose of this paper is to develop a novel multivariate fractional grey model termed GM(a, n) based on the classical GM(1, n) model. The new model can provide accurate prediction with more freedom, and enrich the content of grey theory.Design/methodology/approachThe GM(α, n) model is systematically studied by using the grey modelling technique and the forward difference method. The optimal fractional order a is computed by the genetic algorithm. Meanwhile, a stochastic testing scheme is presented to verify the accuracy of the new GM(a, n) model.FindingsThe recursive expressions of the time response function and the restored values of the presented model are deduced. The GM(1, n), GM(a, 1) and GM(1, 1) models are special cases of the model. Computational results illustrate that the GM(a, n) model provides accurate prediction.Research limitations/implicationsThe GM(a, n) model is used to predict China’s total energy consumption with the raw data from 2006 to 2016. The superiority of the GM(a, n) model is more freedom and better modelling by fractional derivative, which implies its high potential to be used in energy field.Originality/valueIt is the first time to investigate the multivariate fractional grey GM(α, n) model, apply it to study the effects of China’s economic growth and urbanization on energy consumption.

Optimization of the circular microchannel heat sink under viscous heating effect using entropy generation minimization method

The present numerical study carries out the optimization of the channel dimension (DN corresponding to N-number of channels) under the consideration of viscous heating effects in entropy generation (EG) through a uniformly heated microchannel heat sink (MCHS). The numerical computations are performed for a constant total mass flow rate ṁtot and constant total heat flow rateq̇tot. In EG formulation, the partial differential equations (PDEs) are discretized by the first-order forward difference method and the temperature in the denominator is taken as the local one. The EG due to fluid friction (Sgen,FR) and heat conduction (Sgen,HT) individually as well as simultaneously is also reported. The EG due to fluid heat conduction in the radial (Sgen,HT-rad) and axial (Sgen,HT-ax) directions are also examined. The results indicate that the frictional entropy generation (Sgen,FR) is significant towards the micro dimensions of the channel. Additionally, the optimum diameter (D*) corresponding to the optimum number (N*) of the channels is designed at minimum total EG (Sgen,tot).

Using a shifted Grünwald‐Letnikov scheme for the Caputo derivative to study anomalous solute transport in porous medium

SummaryWe propose an extension of the shifted Grünwald‐Letnikov method to solve fractional partial differential equations in the Caputo sense with arbitrary fractional order derivative α and with an advective term. The method uses the relation between Caputo and Riemann‐Liouville definitions, the shifted Grünwald‐Letnikov, and the traditional backward and forward finite difference method. The stability of the method is investigated for the implicit and explicit scheme with homogeneous boundary conditions, and a stability criterion is found for the advective‐dispersive equation. An application of the method is used to solve contaminant diffusion and advective‐dispersive problems. The numerical solution for the fractional diffusion and fractional advection‐dispersion is compared with their respective analytical solutions for different time and space grid refinements. The diffusion simulation exhibited a good fit between the analytical and numerical solutions, with the explicit scheme going from stable to unstable as the time and space refinement changes. The fractional advection‐dispersion application produced small deviations from the analytical solution. These deviations, however, are analogous to the numerical dispersions encountered in conventional finite difference solutions of the advection‐dispersion equation. The new method is also compared with the traditional L2 method. Notably, an example that involves asymmetrical fractional conditions, a fractional diffusivity that depends on time, and a source term show how the methods compare. Overall, this study assesses the quality and easiness of use of the numerical method.

An application of forward difference method in robust stability of discrete uncertainty system with delays

This paper deals with the problems of robust stability for a class of uncertainty parameter systems with delays. By using a new Lyapunov–Krasovskii functional, quadratic inequality, and Schur complement technique, two conditions are developed to guarantee the robust stability of a class of discrete systems with uncertainty parameters in terms of the linear matrix inequality (LMI). By applying the forward difference method and via a quadratic cost function, the criteria of LMI are obtained by the input control u(k)=0 and u(k)=Kx(k); meanwhile, the bounds of cost function are established. The feedback control gain is designed to ensure the robust stability of the closed-loop system. A numerical example and simulation figures are provided to illustrate the effectiveness and potential of the proposed techniques and results.

Simplified algorithm for reliability sensitivity analysis of structures: A spreadsheet implementation.

An important segment of the reliability-based optimization problems is to get access to the sensitivity derivatives. However, since the failure probability is not a closed-form function of the input variables, the derivatives are not explicitly computable and rather require a full reliability analysis which is computationally expensive. In this paper, a step-by-step algorithm has been presented to calculate the derivatives of the probability of failure and safety index with respect to the input parameters based on the advanced first-order second-moment (AFOSM) reliability method. The proposed algorithm is then implemented in a spreadsheet using Visual Basic for Application (VBA) programming language. Two geotechnical and structural examples are then presented to examine the program and describe the modeling procedure. The robustness of the proposed method is examined using a Gaussian random perturbation. The capability of the proposed method in the calculation of the sensitivity derivatives of the model uncertainty is explained in a separate section. Finally, the proposed model has been compared to the forward finite difference (FFD) method and the results are validated.

Dynamic tests and model updating of nonlinear beam structures with bolted joints

Abstract Mechanical joints usually affect structural dynamic responses, leading to local nonlinear stiffness and damping. In this paper, hysteresis Iwan model is discussed in detail to describe nonlinear mechanical behaviors of contact surface. Based on adjusted Iwan model, the derivation process of force-displacement relationship is presented for Iwan beam element. Meanwhile, relevant experimental study on the dynamic behaviors of a cantilever beam with a bolted joint is performed. Linear beam element and Iwan beam element are used to simulate the bolted cantilever beam structure, and the corresponding structural dynamic responses are calculated by forward incremental displacement center difference method. Then, combined with the nonlinear structural model updating method based on the instantaneous characteristics of principal dynamic response components, the bolted cantilever beam structure is updated and the parameters of Iwan beam element are obtained. The updated results are in good agreement with the experimental results, that is, the feasibility and effectiveness of the proposed nonlinear structural model updating method are verified by comparing the updated responses and measured data.

Dynamic ice contact load model for azimuthing thrusters

ABSTRACTFor ships operating in winter conditions, ice loads are important load scenarios for designing azimuthing main propulsion units. One such contact scenario is ice impacting into an azimuthing thruster. In this paper, the formulation of this type of load scenario is presented and existing calculation methods are investigated. A calculation method for dynamic contact load is proposed for ice impact against the main propulsion unit. The basic principle of the proposed load calculation method is an impact of moving bodies combined with dynamic reaction of the structure. The contact load determination is based on the contact pressure–area relationship between an ice block and a steel structure. The dynamic contact problem is solved with a forward difference method. The method is validated with a controlled laboratory-scale test and the test principle is introduced. The dynamic load model is applied to study the effect of dynamics on the response load level.

The Data Fitting and Partial Differential Equation Solving for MQ Quasi-Interpolation

In this paper, the MQ Quasi-interpolation is used to data interpolation, extrapolation and function fitting examples, and compared with the effect of RBF neural network. Using the property of the one-order derivative of MQ Quasi-interpolation, the nonlinear burgers equation is solved by forward difference and Crank-Nicolson finite difference method. The results of numerical experiments verify its convergence and Stability.

Transient Mixed Convection Boundary Layer Flow of an Incompressible Fluid Past a Wedge in Presence of Magnetic Field

In this paper, an analysis is performed to explorethe transient, laminar two-dimensional, mixed convection boundary layer flow of a viscous and incompressible fluid past a vertical wedge taking into account the effect of magnetic field. With appropriate transformations the boundary layer equations are reduced to a local nonsimilarity equations and the solutions are obtained employing three distinct methods, namely, (i) perturbation method for small time; (ii) asymptotic solution method for large time; (iii) straight forward finite difference method for any time. The agreement between the solutions obtained from prescribed methods is found to be excellent. In this study the evaluation of skin-friction coefficient and the local Nusselt number with the effects of different governing parameters such as different time, τ, the exponent, m (= 0.4, 0.5, 1.0), mixed convection parameter, λ (= 0.0, 0.2, 0.4) and magnetic field parameter, M (=0.0, 1.0) for fluids having Prandtl number, Pr= 0.72, 1.0 and 7.0have been discussed. It is observed that both the local skin friction and local Nusseltnumber decreases due to an increase in the value of M. It is also found that an increase in the value of Prandtl number, Pr, leads to a decrease in the value of local skin friction coefficient and the value of local Nusselt number coefficient increases with the increasing values of Prandtl number.