Difference Approximation Research Articles

Higher-order approximations for space-fractional diffusion equation

Second-order and third-order finite difference approximations for fractional derivatives are derived from a recently proposed unified explicit form. The Crank-Nicholson schemes based on these approximations are applied to discretize the space-fractional diffusion equation. We theoretically analyse the convergence and stability of the Crank-Nicholson schemes, proving that they are unconditionally stable. These schemes exhibit unconditional stability and convergence for fractional derivatives of order in the range . Numerical examples further confirm the convergence order and unconditional stability of the approximations, demonstrating their effectiveness in practice.

Heat transfer analysis of Cu-Water nanofluid in a square enclosure using Caputo fractional derivative and machine learning

Fractional-based fluid transport modeling provides a deeper and more comprehensive understanding of the thermophysical properties of small-scale fluids and polymer solutions. Moreover, the fractional-order fluid dynamics framework finds applications in various fields, including renewable energy systems, thermal energy storage, and oil storage tanks. Inspired by these developments, the current study addresses the computational challenges inherent in modeling nanofluid dynamics within a square enclosure by introducing a fractional-order approach to enhance the simulation of buoyant flow and heat transport. Our model integrates the linear interpolation (L1 approach) and finite difference approximation for temporal and spatial derivatives by utilizing the Caputo time-fractional derivative instead of the conventional time derivative. This integration facilitates the use of the Backward Difference Alternating Direction Implicit (BD-ADI) method, which was specifically chosen to reduce the computational burden associated with the non-local characteristics of fractional models. Our results show that by altering the fractional order, significant improvements in heat transfer efficiency are achieved compared to classical models. For instance, compared to the integer order, a lower fractional-order parameter of γ=0.80 with a 4% nanoparticle volume fraction at a Rayleigh number of 103 increases the heat transfer rate by 15.77%. However, at a Rayleigh number of 106, the enhancement reduces to 4.15%, indicating that fractional adjustments’ influence diminishes at higher thermal buoyancy levels. This behavior of the fractional-order nanofluid model, influenced by thermal buoyancy force, has been investigated by using a Multiple Linear Regression(MLR) analysis with 258 numerical samples. Besides, MLR analysis further identifies the nanoparticle volume fraction as a critical factor, influencing the heat transfer rate by 10.48% at lower fractional-order settings.

GPU Accelerating Algorithms for Three-Layered Heat Conduction Simulations

In this paper, we consider the finite difference approximation for a one-dimensional mathematical model of heat conduction in a three-layered solid with interfacial conditions for temperature and heat flux between the layers. The finite difference scheme is unconditionally stable, convergent, and equivalent to the solution of two linear algebraic systems. We evaluate various methods for solving the involved linear systems by analyzing direct and iterative solvers, including GPU-accelerated approaches using CuPy and PyCUDA. We evaluate performance and scalability and contribute to advancing computational techniques for modeling complex physical processes accurately and efficiently.

A Finite Difference-Based Adams-Type Approach for Numerical Solution of Nonlinear Fractional Differential Equations: A Fractional Lotka–Volterra Model as a Case Study

Abstract In this paper, we developed an efficient Adams-type predictor–corrector (PC) approach for the numerical solution of fractional differential equations (FDEs) with a power law kernel. The main idea of the proposed approach is to use a linear approximation to the nonlinear problem and then implement finite difference approximations of derivatives. Numerical comparisons with the fractional Adams method are made and simulation results are demonstrated to evaluate the approximation error of the proposed approach. The efficiency of this approach has been depicted by presenting numerical solutions of some test fractional calculus models. Numerical simulation of a fractional Lotka–Volterra model is provided, as a case study, using the proposed approach. The advantage of the proposed approach lies in its flexibility in providing approximate numerical solutions with high accuracy.

The Impact of a Non-Uniform and Transient Magnetic Field on Natural Convective Jeffrey Fluid Flowing over a Heated Porous Plate: A Comparitive Study

Examining the effects of varying transient magnetic fields and their orientations on boundary layers aids in understanding the flow in complex surface geometries, turbulence control, drag reduction, aircraft and ship design, groundwater management, and the study of construction and coastal engineering. The study introduces a novel dynamics of the boundary layer of a natural convective Jeffrey fluid flowing over an erect porous moving plate under the influence a non-uniform and transient magnetic field M ˜ that grows exponentially over time with α as an accelerating parameter. The governing partial differential equations (PDEs) were non-dimensionalized using similarity variables and then solved analytically using the perturbation technique. The impact of various thermosphysical properties including chemical reactions, thermal source, Jeffrey fluid parameter and plate permeability on the fluid flow were presented graphically utilizing finite difference approximation(FDA) technique in MATLAB ODE15s solver. The study highlighted a notable decrease in shear stress (skin friction) by introducing M ˜ to the vertical plate results in the reduction of momentum boundary layer, accompanied by an increment in thermal boundary layer. Also, the results shows that increasing the magnitude of α leads to decrease the velocity and increase the temperature distribution curve. Additionally, the concentration profile was also found to decrease with stronger chemical reactions. Furthermore, key parameters like heat and mass flux were estimated and discussed through comparison tables. Understanding the effects of unsteady and externally applied increasing magnetic field M ˜ on viscous fluid flow have potential applications in biomedical engineering, such as the design of magnetic drug delivery systems and the analysis of blood flow in human body.

Numerical Simulation of Nonlinear Equations by using Modified Newton Raphson Technique

Non-linear equations are a significant number of complicated issues in mathematics and related subjects that must be solved. The Newton method and its preliminary variations are among the most basic, yet insufficiently effective, methods for resolving non-linear equations. To solve non-linear equations efficiently, a method's desirable feature is to find root with fewer iterations, minimum error (usually less than precision limit), which is typically less than the precision limit. In, this work, we present two approaches (Proposed techniques I and II) for solving non-linear equations by modifying Newton Raphson's technique using first derivative's forward and central difference approximations. Two examples are used to compare performance of the suggested techniques with the existing technique (Secant technique). Finding roots of nonlinear equations under consideration required fewer iterations and a lower absolute relative approximate error while using proposed method II, which performed better than both existing methods (the Secant technique and suggested method I). The suggested approaches I and II were determined to be appropriate substitutes for solving nonlinear equations.

Stability and computational analysis of Influenza-A epidemic model through double time delay

Delay factors demonstration has a significant role in controlling a strain of infectious disease instead of a pharmaceutical strategy. According to the World Health Organization (WHO), 3–5 million cases are reported annually and approximately 290,000 to 650,000 respiratory deaths annually. So, in the present study, we develop a delayed mathematical model based on delay differential equations (DDEs) for the influenza epidemic using a deterministic approach by introducing double delay parameters. The four distinct sub-populations are considered susceptible, exposed, infected, and recovered. For the rigorous analysis, the fundamental properties of the model like positivity, boundedness, existence, and uniqueness, were studied. The influenza-free equilibrium (IFE) and influenza-existing equilibrium (IEE), are the two nonnegative equilibrium points that the model demonstrates. Both locally and globally, the asymptotic stability of the equilibrium points of the model is established and shown under specific situations of reproduction number. Additionally, investigated the model's parameter sensitivity and determined the relative sensitivity of each parameter. Both standard and nonstandard methods—such as Euler, Runge-Kutta, and nonstandard finite difference with a delayed sense—are presented to make computational analysis support a dynamical analysis and the best visualization of results. The stability of the non-standard finite difference scheme is thoroughly analyzed around the steady states of the model. Additionally, the results show that the nonstandard finite difference approximation is an efficient, cost-effective method, independent of time step size, to solve such highly nonlinear and complex real-world problems.

High-Accuracy Positivity-Preserving Finite Difference Approximations of the Chemotaxis Model for Tumor Invasion.

Numerical simulation of the complex evolution process for tumor invasion plays an extremely important role in-depth exploring the bio-taxis phenomena of tumor growth and metastasis. In view of the fact that low-accuracy numerical methods often have large errors and low resolution, very refined grids have to be used if we want to get high-resolution simulating results, which leads to a great deal of computational cost. In this paper, we are committed to developing a class of high-accuracy positivity-preserving finite difference methods to solve the chemotaxis model for tumor invasion. First, two unconditionally stable implicit compact difference schemes for solving the model are proposed; second, the local truncation errors of the new schemes are analyzed, which show that they have second-order accuracy in time and fourth-order accuracy in space; third, based on the proposed schemes, the high-accuracy numerical integration idea of binary functions is employed to structure a linear compact weighting formula that guarantees fourth-order accuracy and nonnegative, and then a positivity-preserving and time-marching algorithm is established; and finally, the accuracy, stability, and positivity-preserving of the proposed methods are verified by several numerical experiments, and the evolution phenomena of tumor invasion over time are numerically simulated and analyzed.

A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods.

Discrete time stability of augmented Lagrangian formalism based underactuated inverse dynamics control method

Stability problems of robotic systems arise sometimes suddenly, seemingly for no reason. The digital time sampling is often the main cause of these instabilities. Discrete models, which are capable of the prediction of stability, are available for low-degree-of-freedom template models of linear position and force control. However, for the inverse dynamics control of underactuated systems, the literature has a lack of generally applicable results related to the effect of time discretization. Several control approaches are available in the literature out of which a widely used one, the augmented Lagrangian formalism and its stability properties are analysed in this work. Theoretical stability properties are obtained for a generally usable, linear, underactuated, two-degree-of-freedom constrained template model. The actuator dynamics, the finite difference approximation of the feedback velocity and the filtering of the feedback data are considered in the model. These phenomena strongly affects the stability properties. The theoretically obtained stability maps are experimentally validated on an underactuated crane-like indoor robot. The position and orientation accuracy of the robot were assessed: the absolute position error was below 30 mm and the orientation error was below 3°.

Semi-analytical solutions for steady-state bending-bending coupled forced vibrations of a rotating wind turbine blade by means of Green's functions

Wind power has received significant attention in recent years as a renewable energy source. Wind turbine blades, characterized by their long lengths, are prone to damage as a result of aeroelastic instability. This paper aims to derive semi-analytical solutions for steady-state forced vibrations of rotating wind turbine blades, considering the coupling of bending, bending, and unsteady aerodynamic loads. The novelty of this work lies in the use of the Green's function method to solve differential dynamic equations with variable coefficients, which are induced by aerodynamic forces. To model a wind turbine blade, the Euler-Bernoulli beam model is employed, and Greenberg's expressions are taken into consideration. Governing equations for coupled vibrations of the blade are then determined. In order to solve the governing equations, displacement solutions are decomposed into quasi-static displacements and dynamic displacements. The Laplace transformation and Green's function methods are utilized to obtain fundamental solutions of the governing equations. Subsequently, employing the principle of superposition, Fredholm integral equations for steady-state forced vibration of a rotating wind turbine blade are derived. To spatially discretize Fredholm integral equations, compound trapezoid formulae and central finite difference approximations are employed. This discretization process leads to the formation of a system of algebraic equations. Semi-analytical solutions for coupled forced vibrations of the rotating wind turbine blade are obtained by solving the algebraic equations. In the numerical solution part, validation of proposed solutions are verified by comparing them with numerical and finite element solutions in the literature. Influences of some important physical parameters, such as the rotating velocity, the setting angle, the cone angle, the inflow ratio, and damping, on vibration responses of blades are discussed.

A Spatial Reconstruction Method of Ionospheric foF2 Based on High Accuracy Surface Modeling Theory

The ionospheric F2 critical frequency (foF2) is one of the most crucial application parameters in high-frequency communication, detection, and electronic warfare. To improve the accuracy of spatial reconstruction of the ionospheric foF2, we propose a high-accuracy surface (HAS) modeling method. This method converts difficult-to-solve differential equations into more manageable algebraic equations using direct difference approximation, significantly reducing algorithm complexity and computational load while exhibiting excellent convergence properties. We used seven stations in Brisbane, Canberra, Darwin, Hobart, Learmonth, Perth, and Townsville, with one station as a validation station and six as training stations (e.g., Brisbane as a validation station and the other stations—Canberra, Darwin, Hobart, Learmonth, Perth, and Townsville—as training stations). The training stations and the HAS method were used to train and reconstruct the validation stations at different solar activity periods, seasons, and local times. The predicted values of the validation stations were compared with the measured values, and the proposed method was analyzed and validated. The reconstruction results show the following. (1) The relative root mean square errors (RRMSEs) of HAS method prediction in different solar activity epochs were 13.67%, 7.74%, and 9.19%, respectively, which are 13.57%, 7.41%, and 6.41% higher than the prediction accuracy of the Kriging method, respectively. (2) In the four seasons, the RRMSEs of the HAS method prediction are 9.27%, 13.1%, 8.81%, and 8.09%, respectively, which are 10.83%, 11.73%, 4.25%, and 12.00% higher than the prediction accuracy of the Kriging method. (c) During the daytime and nighttime, the RRMSEs of HAS method prediction were 9.23% and 11.17%, which were 5.92% and 11.99% higher than the prediction accuracy of the Kriging method, respectively. (d) Under the validation dataset, the average predictive RRMSE of the HAS method was 10.29%, and the average predictive RRMSE of the IRI prediction model was 12.35%, with a 2.06% improvement in the predictive accuracy of the HAS method. In general, the prediction effect of the HAS method was better than that of the Kriging method, thus verifying the effectiveness and reliability of the proposed method. In summary, the proposed reconstruction method is of great significance for improving usable frequency prediction and enhancing communication performance.

Machine-learning aided calibration and analysis of porous media CFD models used for rotating packed beds

The proposed research is an attempt to advance the state-of-the-art of the numerical modelling of RPB by combining Computational Fluid Dynamics and Machine Learning approaches. The latter creates an accurate framework that should help quantify the potential of Rotating Packed Beds (RPB) technology to intensify conventional CO2 capture processes. To this end, a direct sensitivity analysis is detailed to supplement a machine-learning (ML) algorithm built for calibrating resistance coefficients needed for porous media modelling. The algorithm is used to improve CFD predictions of dry pressure drop in rotating packed beds (RPBs) for a wide range of operating conditions. The sensitivity derivatives with respect to the packing resistance coefficients are demonstrated for the first time in RPBs, which is not available in the current CFD open source and commercial codes. In this regard, sensitivity differential equations are derived from three-dimensional Navier-Stokes equations for porous media in a rotating reference frame. These sensitivity equations are discretized using a finite volume scheme and solved to obtain the sensitivity pressure drop differences at the packing edges. The results are validated against the predictions of the analytical sensitivity analysis and the finite difference approximation. After, the Newton – Gauss method that employs the sensitivity pressure drop derivatives, is used to minimize the error (cost function) between the pressure drop obtained from CFD simulations and the available experimental data. This is achieved by tuning the packing resistance coefficients to the RBPs' operating conditions (gas flowrate and rotating speed) and correlate them using an artificial neural network (ANN). The results of the proposed approach show a significant improvement in porous media-based CFD predictions of RPBs' pressure drop across a wide range of operating conditions and this over conventional porous media-based CFD models. This is necessary for CFD models to be reliably used as a tool that can efficiently improve existing RPBs' designs and/or participate in RPBs' design innovation.

THE PREDICTOR-CORRECTOR BACKWARD DIFFERENCE FORMULAS TO ESTIMATE THE TREND OF RIVER WATER QUALITY

The time-series trend analysis is a mathematical technique that attempts to estimate future data movement based on observed previous and current data. This can include analysing long-terms changes in river water quality in order to emphasise river management in the future. Therefore, in this study, the backward difference formula (BDF) with forward and central difference approximations is derived to estimate a of 20-year trend in water quality index (WQI) data at the Selangor River basin in Malaysia. As an adaptation of the conventional BDF and predictor-corrector scheme, the Euler’s method with a central difference formula is introduced as a predictor formula to forecast future values. Subsequently, BDF with central and forward difference formulas are specifically designed as corrector formulas to adjust the predicted future values. The performance of the method is evaluated with absolute error and average error accuracy measure indicators. The numerical results validated the proposed method and proved its reliability in estimating the time series trend analysis owing to its reasonable accuracy. In conclusion, the outcomes of this study can provide the authorities with a technical reference for monitoring, maintaining and improving the river water quality in Selangor and other states.

Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions

Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions

Approximation of one and two dimensional nonlinear generalized Benjamin-Bona-Mahony Burgers' equation with local fractional derivative

This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative. The unknown solution of the differential problem and corresponding derivatives are estimated using Haar wavelet approximations (HWA). The collocation procedure is then implemented in HWA, to transform the given model to the system of linear algebraic equations for the determination of unknown constant coefficient of the Haar wavelet series, which update the derivatives and the numerical solutions. The sufficient condition is established for the stability of the proposed technique, and then verified computationally. To check the performance of the scheme, few illustrative examples in one and two dimensions along with l∞ and l2 error norms are also given. Besides this, the computational convergence rate is calculated for both type equations. Additionally, computed solutions are compared with available results in literature. Simulations and graphical data discloses, that suggested scheme works well for such complex problems.

Numerical Finite-Difference Approximations of a Coupled Reaction-Diffusion System with Gradient Terms

This study focuses on the derivation of explicit and implicit finite difference formulas.The objective of this study is to derive an estimation of the blow-up time for a coupled reaction-diffusion system incorporating gradient terms, employing numerical finite difference approximations. Furthermore, an examination is conducted on the consistency, stability, and convergence of the proposed schemes. Additionally, the study presents two numerical experiments. In each instance, the numerical blow-up time is calculated benefit the suggested methodologies, employing varying space steps and non-fixed time-stepping. The numerical findings obtained demonstrate that the blow-up time sequence exhibits convergence as the space step decreases. Moreover, thenumerical orders of convergence for the blow-up time goes well with the theoretical orders observed in the numerical solutions.

Multiscale model reduction for the time fractional thermoporoelasticity problem in fractured and heterogeneous media

In this paper, we consider the time fractional thermoporoelasticity problem in fractured and heterogeneous media. The mathematical model with a time memory formalism is described by a coupled system of equations for pressure, temperature and displacements. We use an implicit finite difference approximation for temporal discretization. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) for two-dimensional model problems. Further, we use the Generalized Multiscale Finite Element Method (GMsFEM) for coarse grid approximation. The primary concept behind the proposed method is to streamline the complexity inherent in the thermoporoelasticity problem. Given that our model equation incorporates multiple fractional powers, leading to multiple unknowns with memory effects, we aim to address this intricacy by optimizing the problem’s dimensionality. As a result, the solution is sought on a coarse grid, a strategic choice that not only simplifies the computational cost but also contributes to significant time savings. We present numerical results for the two-dimensional model problems in heterogeneous fractured porous media. We derive relative errors between the reference fine grid solution and the multiscale solution for different numbers of multiscale basis functions. The results confirm that the proposed method is able to achieve good accuracy with a few degrees of freedoms on the coarse grid.

Numerical integration method for two-parameter singularly perturbed time delay parabolic problem

This study presents an (ε, μ)−uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank–Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's 1/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature.

Structural Design and Parameter Optimization of Magnetic Gradient Tensor Measurement System.

Magnetic anomaly detection (MAD) technology based on the magnetic gradient tensor (MGT) has broad application prospects in fields such as unexploded ordnance detection and mineral exploration. The difference approximation method currently employed in the MGT measurement system introduces measurement errors. Designing reasonable geometric structures and configuring optimal structural parameters can effectively reduce measurement errors. Based on research into differential MGT measurement, this paper proposes three simplified planar MGT measurement structures and provides the differential measurement matrix. The factors that affect the design of the baseline distance of the MGT measurement system are also theoretically analyzed. Then, using the magnetic dipole model, the error analysis of the MGT measurement structures is carried out. The results demonstrate that the planar cross-shaped structure is optimal, with the smallest measurement error, only 3.15 × 10-10 T/m. Furthermore, employing the control variable method, the impact of sensor resolution constraints, noise level, target magnetic moment, and detection distance on the design of the optimal baseline distance of the MGT measurement system is simulated and verified. The results indicate that the smaller the target magnetic moment, the farther the detection distance, the lower the magnetometer resolution, the greater the noise, and the greater the baseline distance required. These conclusions provide reference and guidance for the construction of the MGT measurement system based on triaxial magnetometers.