Backward Finite Difference Research Articles

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

Kinematics of nonlinear waves over variable bathymetry. Part I: Numerical modelling, verification and validation

Fluid particle kinematics due to wave motion (i.e. orbital velocities and accelerations) at and beneath the free surface is involved in many coastal and ocean engineering applications, e.g. estimation of wave-induced forces on structures, sediment transport, etc. This work presents the formulations of these kinematics fields within a fully nonlinear potential flow (FNPF) approach. In this model, the velocity potential is approximated with a high-order polynomial expansion over the water column using an orthogonal basis of Chebyshev polynomials of the first kind. Using the same basis, original analytical expressions of the components of velocity and acceleration are derived in this work. The estimation of particle accelerations in the course of the simulation involves the time derivatives of the decomposition coefficients, which are computed with a high-order backward finite-difference scheme in time. The capability of the numerical model in computing the particle kinematics is first validated for regular nonlinear waves propagating over a flat bottom. The model is shown to be able to predict both the velocity and acceleration of highly nonlinear and nearly breaking waves with negligible error compared to the corresponding stream function wave solution. Then, for regular waves propagating over an uneven bottom (bar-type bottom profile), the simulated results are confronted with existing experimental data, and very good agreement is achieved up to the sixth-order harmonics for free surface elevation, velocity and acceleration.

A novel dynamic predictive control model for variable air volume air conditioning systems using deep learning

Model-based predictive control has proven effective for managing indoor temperature and humidity in variable air volume (VAV) air conditioning systems. However, delays in temperature and humidity adjustments in response to changes in internal and external environmental conditions can impede high-performance operation. This study introduces a dynamic predictive control model for the multi-zone indoor temperature and humidity of VAV systems, designed to predict and control these parameters in real-time. Utilizing the RC network two-node wall structure method and backward finite difference scheme, a multi-zone building model is developed. Coupled with a deep fuzzy cognitive map (DFCM) for temperature and humidity prediction, and controlled by a PID controller, the model dynamically regulates the opening degrees of terminal air dampers and chilled water valves. Implemented on a networked control platform in a large commercial building, the model consistently maintained indoor temperature and humidity within ±0.5 °C and ±0.1 g/(kg dry air) of the target values under varying conditions, demonstrating substantial improvements in stability and operational efficiency. The research enhances the predictability and control of HVAC systems through advanced algorithmic integration, improving real-time indoor climate management in complex building environments. This method offers a practical improvement in HVAC control technologies, which could be beneficial for applications in commercial building management systems, providing a useful tool for supporting energy efficiency and sustainability in modern infrastructures.

Multiaxial ratcheting-fatigue evaluation of pressurized elbow pipe under strong cyclic loading using damage-coupled cyclic plasticity model

Continuous damage mechanics (CDM) is based on the theories of continuous medium mechanics and continuous medium thermodynamics, which considers damage is an irreversible dissipative process within the materials, and employs the field-theoretic approach of image-only science to study the internal macroscopic damage evolution law of the materials and its influence on the deterioration of the macroscopic mechanical properties of the materials. Hence, within the framework of CDM, a damage-coupled cyclic plasticity constitutive model is proposed based on combined isotropic and Chen-Jiao-Kim (CJK) kinematic hardening rule for evaluating the ratcheting-fatigue behavior of 90° elbow pipes under strong cyclic loading in this paper. Stress return mapping and numerical solution procedures of the proposed model are formulated based on the backward finite difference method. Formulation of the consistent tangent operator is presented for the Finite Element implementation of the plasticity model in ABAQUS UMAT. The FE analysis of non-pressurized and pressurized elbow pipes under cyclic displacement-controlled loading is respectively performed using the implemented constitutive model. The results reveal that the predicted results of the damage-coupled cyclic plasticity constitutive model are better than combined isotropic-kinematic hardening model, and well consistent with the experimental data.

SEVERE CYCLONIC STORM SITRANG: A METEOROLOGICAL ANALYSIS OF ITS FORMATION, INTENSITY, AND IMPACT ON BANGLADESH

Employing a non-central difference method of lines in tandem with the 4th order Runge-Kutta technique, this study adopts a sophisticated computational approach, ensuring precision and efficiency to resolve the shallow water equations and predict water levels caused by a cyclone in the coastal area of Bangladesh. To discretize spatial derivatives, a 4-point backward finite difference method utilized while keeping time derivatives continuous. The authors transformed the shallow water equations with boundary conditions into an initial value problem and used the Runge-Kutta(4,4) method for solving this transformed initial value problem. To effectively include the land-sea boundary and bottom structure at a reasonable charge, the authors used a high-resolution approach. To create a regular and stable tidal oscillation in the area, the authors applied the tidal component to the southern boundary of the primary model. Then ran the surge model to estimate water elevations caused by the nonlinear interaction between tide and surge. Using numerical experiments, the authors simulated water heights generated by tide, pure surge, and tide surge interaction caused by the intense cyclonic storm SITRANG. The study's simulated results demonstrate a commendable alignment with the reported data.

On the BEM solution of convection–diffusion type equations involving variable convective coefficients

In this study, numerical solutions of the unsteady convection–diffusion type equations of variable convective coefficients are investigated by two effective techniques alternative to direct boundary element method (BEM). That is, the domain boundary element and the dual reciprocity BEMs with the fundamental solution of convection–diffusion equation are used in space to transform the governing differential equations into equivalent integral equations while a backward finite difference scheme is utilized in time discretization. The fruitfulness of these combined techniques are shown by the implementation of the techniques for some widely investigated fluid dynamics problems. In fact, the lid-driven cavity flow is solved, and precise results are attained as benchmarks for assessing the accuracy of the aforementioned methods. Further, the application of the methods is extended for the natural convection flow which is governed mainly by the convection–diffusion type equations accompanied by the velocity components as variable convective coefficients. The obtained numerical results reveal that the domain BEM which uses the fundamental solution of convection–diffusion equation captures the characteristics and physical advancement of the fluid flow quite well at various combined values of the problem physical parameters.

Graded mesh modified backward finite difference method for two parameters singularly perturbed second-order boundary value problems

The existence of boundary layers in the solutions of two-parameter singularly perturbed boundary value problems makes classical numerical methods insufficient in providing accurate approximations. Consequently, the development of layer-adapted mesh methods that achieve parameter uniform convergence and are specifically designed to accurately handle these layers becomes highly significant. The aim of this paper is to construct and examine a numerical approach for obtaining approximate solutions to a specific class of two-parameter singularly perturbed second order boundary value problems whose solutions exhibit boundary layers at both ends of the domain. The problem is discretized by employing a modified backward finite difference method on a mesh that is graded. To validate the theoretical findings, well known test problems from the existing literature are utilized. Moreover, the efficiency of the proposed method is demonstrated by comparing it to other existing methods in the literature. The stability and uniform convergence with respect to the parameters of the proposed method have been verified, revealing that it attains second-order convergence in the maximum norm. The numerical outcomes illustrate that the proposed method offers remarkably accurate approximations of the solution. Theoretical findings are in agreement with the experimental results.

Time-dependent simplified [formula omitted] approximation for neutron transport equation

Time-dependent transport is still too computationally expensive for various practical purposes. On the other hand, the diffusion approximation is unable to provide accurate results for many real problems. To establish a middle ground between the accuracy of the transport and the simplicity of the diffusion approximations, the simplified spherical harmonics (SPN) method was introduced. The present paper aims to improve the accuracy of the time-dependent SP3 method by employing the simplified double-P1 (SDP1) approximation, which is based on the one-dimensional double-spherical harmonics DP1 equations. Using the finite element method to treat the spatial dependence of the SDP1 equations leads to a set of first-order ordinary differential equations on the time variable. These equations are then converted into a set of linear algebraic equations by discretizing the time variable with the implicit backward finite difference scheme. Our numerical results indicate that the SDP1 approximation could compute more precise values than the widely used SP3 method in transient problems.

A New Efficient Hybrid Method Based on FEM and FDM for Solving Burgers’ Equation with Forcing Term

This paper presents a study on the numerical solutions of the Burgers’ equation with forcing effects. The article proposes three hybrid methods that combine two‐point, three‐point, and four‐point discretization in time with the Galerkin finite element method in space (TDFEM2, TDFEM3, and TDFEM4). These methods use backward finite difference in time and the finite element method in space to solve the Burgers’ equation. The resulting system of the nonlinear ordinary differential equations is then solved using MATLAB computer codes at each time step. To check the efficiency and accuracy, a comparison between the three methods is carried out by considering the three Burgers’ problems. The accuracy of the methods is expressed in terms of the error norms. The combined methods are advantageous for small viscosity and can produce highly accurate solutions in a shorter time compared to existing numerical schemes in the literature. In contrast to many existing numerical schemes in the literature developed to solve Burgers’ equation, the methods can exhibit the correct physical behavior for very small values of viscosity. It has been demonstrated that the TDFEM2, TDFEM3, and TDFEM4 can be competitive numerical methods for addressing Burgers‐type parabolic partial differential equations arising in various fields of science and engineering.

Artificial intelligence computing analysis of fractional order COVID-19 epidemic model

Artificial intelligence plays a very prominent role in many fields, and of late, this term has been gaining much more popularity due to recent advances in machine learning. Machine learning is a sphere of artificial intelligence where machines are responsible for doing daily chores and are believed to be more intelligent than humans. Furthermore, artificial intelligence is significant in behavioral, social, physical, and biological engineering, biomathematical sciences, and many more disciplines. Fractional-order modeling of a real-world problem is a powerful tool for understanding the dynamics of the problem. In this study, an investigation into a fractional-order epidemic model of the novel coronavirus (COVID-19) is presented using intelligent computing through Bayesian-regularization backpropagation networks (BRBFNs). The designed BRBFNs are exploited to predict the transmission dynamics of COVID-19 disease by taking the dataset from a fractional numerical method based on the Grünwald–Letnikov backward finite difference. The datasets for the fractional-order mathematical model of COVID-19 for Wuhan and Karachi metropolitan cities are trained with BRBFNs for biased and unbiased input and target values. The proposed technique (BRBFNs) is implemented to estimate the integer and fractional-order COVID-19 spread dynamics. Its reliability, effectiveness, and validation are verified through consistently achieved accuracy metrics that depend on error histograms, regression studies, and mean squared error.

Fractional-Step Method with Interpolation for Solving a System of First-Order 2D Hyperbolic Delay Differential Equations

In this article, we consider a delayed system of first-order hyperbolic differential equations. The presence of the delay term in first-order hyperbolic delay differential equations poses significant challenges in both analysis and numerical solutions. The delay term also makes it more difficult to use standard numerical methods for solving differential equations, as these methods often require that the differential equation be evaluated at the current time step. To overcome these challenges, specialized numerical methods and analytical techniques have been developed for solving first-order hyperbolic delay differential equations. We investigated and presented analytical results, such as the maximum principle and stability results. The propagation of discontinuities in the solution was also discussed, providing a framework for understanding its behavior. We presented a fractional-step method using a backward finite difference scheme and showed that the scheme is almost first-order convergent in space and time through the derivation of the error estimate. Additionally, we demonstrated an application of the proposed method to the problem of variable delay differential equations. We demonstrated the practical application of the proposed method to solving variable delay differential equations. The proposed algorithm is based on a numerical approximation method that utilizes a finite difference scheme to discretize the differential equation. We validated our theoretical results through numerical experiments.

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.

A high-order convergence analysis for semi-Lagrangian scheme of the Burgers' equation

<abstract><p>In this article, we provide a comprehensive convergence and stability analysis of a semi-Lagrangian scheme for solving nonlinear Burgers' equations with a high-order spatial discretization. The analysis is for the iteration-free semi-Lagrangian scheme comprising the second-order backward finite difference formula (BDF2) for total derivative and the fourth-order central finite difference for diffusion term along the trajectory. The main highlight of the study is to thoroughly analyze the order of convergence of the discrete $ \ell_2 $-norm error $ \mathcal{O}(h^2+\triangle x^4+ \triangle x^{p+1}/h) $ by managing the relationship between the local truncation errors from each discretization procedure and the interpolation properties with a symmetric high-order discretization of the diffusion term. Furthermore, stability is established by the uniform boundedness of the numerical solution using the discrete Grönwall's Lemma. We provide numerical examples to support the validity of the theoretical convergence and stability analysis for the propounded backward semi-Lagrangian scheme.</p></abstract>

A SHOOTING METHOD SOLUTION FOR A CONVECTIVE-RADIATIVE STRAIGHT LONGITUDINAL FIN WITH VARIABLE THERMAL PROPERTIES AND NONLINEAR BOUNDARY CONDITIONS

We develop in this work a simple and highly efficient shooting approach for solving the fin energy equation with multiple nonlinearities. The present fin problem is characterized by temperature-dependent thermal conductivity, heat transfer coefficient, and surface radiation emissivity, where the fin base is imposed to a constant temperature and the fin tip is subjected to a combination of convective and radiative heat losses. The governing fin boundary value problem is first reduced into an equivalent initial value problem and then integrated using the fourth-order Runge-Kutta method. The temperature gradient at the tip is approximated by a five-point backward finite difference formula, and computed iteratively using the secant method on the base, which is decisive for numerical integration. The fin problem is solved and compared for two cases of tip boundary condition: an adiabatic fin tip and a convective-radiative fin tip. A thermal analysis is performed using Biot number, Stark number, and the geometrical number that stands for the ratio of fin surface area to its cross-sectional area. Solutions are computed and compared to those obtained by the boundary value problem method, Galerkin method, and the Adomian decomposition method under the assumption of adiabatic fin tip. The accuracy of the nonlinear shooting method is checked by evaluating the absolute errors. Comparative results show that the fin temperature distribution and the fin efficiency are significantly impacted not only by Biot number, Stark number, and geometrical number but also by the type of the tip boundary condition, which could dramatically degrade the fin efficiency mainly for small values of geometrical number.

On a moving boundary problem associated with the swelling drug release platforms

This paper studies a parabolic-type system of time-dependent partial differential equations coupled with Robin boundary conditions to model the solvent penetration and subsequently a controlled drug release from a swellable polymeric spherical platform. The model's key feature is that the solvent penetration and the polymer swelling result in moving boundaries. From an analytical perspective, some properties of the solutions, such as the positivity and the uniqueness of the solutions, are established. A second-order iterative procedure based on the backward finite difference method is carried out to solve the problem numerically. Finally, the numerical results and the asymptotic solutions are compared, intending to illustrate the validity and applicability of the numerical scheme.

Adsorptive removal of Pb(II) using nanostructured γ-alumina in a packed bed adsorber: Simulation using gPROMS.

In this work, convective-dispersive and pore volume and surface diffusion models have been used to analyze Pb(II) adsorption from an aqueous solution over a nanostructured γ-alumina adsorbent in a packed bed adsorber. The models encompassing partial differential equation and a linear algebraic equation coupled with isotherm have been simulated in gPROMS using the backward finite difference approach. The predicted breakthrough curves of Pb(II) adsorption concerning flow rate, initial metal concentration, and bed height were matched with the experimental data. The accuracy of model predictions was analyzed through statistical measures such as coefficient of determination (R2), root mean square error, and chi-squared value. The simulation results also predicted the axial dispersion, distribution coefficient, mass transfer coefficient, pore volume, and surface diffusion coefficient, which are, otherwise, difficult to measure experimentally and, in turn, have been used to assess the mass transfer characteristics of continuous Pb(II) adsorption. Additionally, the values of breakthrough time, exhaustion time, adsorption column capacity, and mass transfer zone were determined as a function of flow rate, bed height, and initial metal concentration. Surface and pore volume diffusions (10-11-10-10 m2/s) apparently controlled the continuous adsorption process, with surface diffusion being dominant. The transport parameters evaluated in the current study could be beneficial for the large-scale Pb(II)/nanostructured γ-alumina adsorption system. As evident from the successful simulation, the developed gPROMS program can also be applied to other adsorbate/adsorbent systems with a slight modification concerning the operating parameters.

Analysis of a Mathematical Model Arising in Plant Disease Epidemiology

In this work we study from the mathematical and numerical point of view a problem arising in vector-borne plant diseases. The model is written as a nonlinear system composed of a parabolic partial differential equation for the vector abundance function and a first-order ordinary differential equation for the plant health function. An existence and uniqueness result is proved using backward finite differences, uniform estimates and passing to the limit. The regularity of the solution is also obtained. Then, using the finite element method and the implicit Euler scheme, fully discrete approximations are introduced. A discrete stability property and a main a priori error estimates result are proved using a discrete version of Gronwall’s lemma and some estimates on the different approaches. Finally, some numerical results, in one and two dimensions, are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.

Numerical Solution of Linear Fractional Differential Equation with Delay Through Finite Difference Method

This article addresses a new numerical method to find a numerical solution of the linear delay differential equation of fractional order , the fractional derivatives described in the Caputo sense. The new approach is to approximating second and third derivatives. A backward finite difference method is used. Besides, the composite Trapezoidal rule is used in the Caputo definition to match the integral term. The accuracy and convergence of the prescribed technique are explained. The results are shown through numerical examples.

An estimation of water levels associated with a storm along the coast of Bangladesh using a non-central difference method of lines

In this paper, depth-averaged shallow water equations were solved using a non-central difference method of lines together with the classical RK(4,4) method to estimate water levels associated with a cyclone along the coast of Bangladesh. A 4-point backward finite difference method was adopted in discretizing spatial derivatives retaining time derivatives as continuous. Thus, the shallow water equations together with the boundary conditions were transformed into a system of initial value problems, which was solved using the RK(4,4) method. A nesting approach was employed to incorporate the land-sea interface and bottom topography effectively with economic cost. A stable tidal oscillation was generated in the domain applying the M2 tidal constituent along the southern boundary of the parent model and the surge model was then run to get water levels owing to dynamic tide-surge interaction. Numerical experiments were performed to simulate water levels due to tide, surge, and nonlinear tide-surge interactions associated with the cyclonic storm, AILA. The obtained results were in good agreement with those attained in some previous studies and observations and the WLs due to the nonlinear tide-surge interaction were in a reasonable agreement with respect to the root mean square error values and predictive skills.

Large strain Flory’s decomposition for Lagrangian modeling of viscoleastic solids and compressive fluids

The fundamental difference in the solution of solids and fluids relies on the respective constitutive laws. Based on the Rivlin-Saunders-Düster-Hartmann hyperelastic model and using the Flory’s strain decomposition, we present a new total Lagrangian viscoelastic constitutive model for both Kelvin-Voigt viscoelastic solids and free-surface compressive viscous isothermal fluids. A dissipative viscous virtual work is written as a function of the time rate of isochoric invariants and its relation with the viscous stress is derived. Local time derivatives are solved by backward finite difference, allowing a consistent tangent viscoelastic constitutive tensor. The virtual work principle is used to write the weak equilibrium equation and its position-based finite element counterpart. Dynamic time integration is carried out by the Newmark β method and the Newton-Raphson procedure is used to solve time steps. The formulation is validated against experimental and numerical literature results revealing good precision. Additional examples are shown in order to demonstrate the applicability and future possibilities of the technique.