Explicit Finite Difference Approximation Research Articles

HNS: An efficient hermite neural solver for solving time-fractional partial differential equations

Neural network solvers represent an innovative and promising approach for tackling time-fractional partial differential equations by utilizing deep learning techniques. L1 interpolation approximation serves as the standard method for addressing time-fractional derivatives within neural network solvers. However, we have discovered that neural network solvers based on L1 interpolation approximation are unable to fully exploit the benefits of neural networks, and the accuracy of these models is constrained to interpolation errors. In this paper, we present the high-precision Hermite Neural Solver (HNS) for solving time-fractional partial differential equations. Specifically, we first construct a high-order explicit approximation scheme for fractional derivatives using Hermite interpolation techniques, and rigorously analyze its approximation accuracy. Afterward, taking into account the infinitely differentiable properties of deep neural networks, we integrate the high-order Hermite interpolation explicit approximation scheme with deep neural networks to propose the HNS. The experimental results show that HNS achieves higher accuracy than methods based on the L1 scheme for both forward and inverse problems, as well as in high-dimensional scenarios. This indicates that HNS has significantly improved accuracy and flexibility compared to existing L1-based methods, and has overcome the limitations of explicit finite difference approximation methods that are often constrained to function value interpolation. As a result, the HNS is not a simple combination of numerical computing methods and neural networks, but rather achieves a complementary and mutually reinforcing advantages of both approaches. The data and code can be found at https://github.com/hsbhc/HNS.

Simulation tumor growth in heterogeneous medium based on diffusion equation

In this paper, the diffusion equation is used to model the spatio-temporal dynamics of a tumor, taking into account the heterogeneity of the medium. This approach allows us to take into account the complex geometric shape of the tumor when modeling. The main purpose of the work is to demonstrate the applicability of this approach by comparing the results obtained with the data from clinical observations. We use an algorithm based on an explicit finite-difference approximation of differential operators to solve the diffusion equation. The ranges of possible values that can take the input parameters of the model to match the results of clinical observations are obtained. On the basis of the data of clinical observations, the relative error of the results of computational experiments was determined, which lies in the range from 1.8% to 14.6%. It is concluded that the heterogeneity of the physical parameters of the model, in particular the diffusion coefficient, has a significant effect on the shape of the tumor.

Building of a Mathematical Model for Solving the Elastic–Gas-Dynamic Task of the Gas Lubrication Theory for Petal Bearings

Petal gas bearings are widely used in industrial turbine installations. However, there are practically no mathematical models that take into account the nonlinear dependence of the elastic element deformations of the gas-dynamic bearing on the pressure. It is very difficult to obtain a converged solution with a given accuracy in an acceptable time. On this basis, a mathematical model of the motion of a gas-dynamic bearing with a petals package was proposed. The model takes into account the pressure dependence of the elastic element deflections of the bearing, as well as the bearing design features and its elastic elements. A feature of solving the presented task was the joint solution of two subtasks: gas lubrication and deformations of the elastic elements of the bearing. The problem was that the contact areas of the elastic elements are not known in advance. The search for deflections or the formulation of the elasticity problem was based on the Lagrange variational principle. Petals are in the shape of thin cylindrical shells. The solution was achieved by minimizing the potential energy of the system of deformed petal shells using the first-order gradient method. The solution to the gas-dynamics problem was achieved by applying an explicit finite-difference approximation in time. To implement the proposed model, a numerical algorithm was developed to solve the related tasks of dynamics and elasticity by combining the methods of variational calculus, optimization, and an explicit finite-difference time scheme. The order of accuracy of convergence of the solution corresponded to 10−5. The study demonstrated that the load-carrying capability of the bearing increased by 2–4 times with an increase in the number of petals. The results of experimental studies allowed us to estimate the interval for the ascent speed of the rotor, which was used as the initial conditions for numerical modeling.

Solving a System of Caputo Fractional-Order Volterra Integro-Differential Equations with Variable Coefficients Based on the Finite Difference Approximation via the Block-by-Block Method

This paper focuses on computational technique to solve linear systems of Volterra integro-fractional differential equations (LSVIFDEs) in the Caputo sense for all fractional order linsin0,1 using two and three order block-by-block approach with explicit finite difference approximation. With this method, we aim to use an appropriate process to transform our problem into an analogous piecewise iterative linear algebraic system. Moreover, algorithms for treating LSVIFDEs using the above process have been developed, in order to express these solutions. In addition, numerical examples for our scheme are presented based on various kernels, including symmetry kernel and other sorts of separate kernels, are used to illustrate the validity, effectiveness and applicability of the suggested approach. Consequently, comparisons are performed with exact results using this technique, to communicate these approaches most general programs are written in Python V 3.8.8 software 2021.

Solving a System of Fractional-Order Volterra Integro-Differential Equations Based on the Explicit Finite Difference Approximation via the Trapezoid Method with Error Analysis

The well-known central finite difference approximation was combined with the trapezoid quadrature method in this study to provide a numerical solution of the linear system of Volterra integro-fractional differential equations (LSVI-FDEs) of arbitrary orders, where the fractional derivative is described in the Caputo sense and the orders are between zero and one. The method works by first using the central finite difference approximation to approximate the Caputo derivative at any fixed point and then using the trapezoidal rule to obtain a finite difference expression for our fractional equation, while recalling the linear spline approximation for the first steps. This new, more efficient method involves converting sets of equations and conditions into matrix relationships, from which symmetry matrices can be created in some cases. We also present a new approach for error analysis of the discrete numerical scheme and the explicit numerical technique for LSVI-FDEs. The multi-level explicit finite difference approximation’s stability and convergence were explored, and a MatLab application was created to explain the results. Finally, several numerical examples are offered to demonstrate the technique’s application.

Fast hybrid explicit group methods for solving 2D fractional advection-diffusion equation

<abstract><p>In recent years, fractional partial differential equations (FPDEs) have been viewed as powerful mathematical tools for describing ample phenomena in various scientific disciplines and have been extensively researched. In this article, the hybrid explicit group (HEG) method and the modified hybrid explicit group (MHEG) method are proposed to solve the 2D advection-diffusion problem involving fractional-order derivative of Caputo-type in the temporal direction. The considered problem models transport processes occurring in real-world complex systems. The hybrid grouping methods are developed based upon a Laplace transformation technique with a pair of explicit group finite difference approximations constructed on different grid spacings. The proposed methods are beneficial in reducing the computational burden resulting from the nonlocality of fractional-order differential operator. The theoretical investigation of stability and convergence properties is conducted by utilizing the matrix norm analysis. The improved performance of the proposed methods against a recent competitive method in terms of central processing unit (CPU) time, iterations number and computational cost is illustrated by several numerical experiments.</p></abstract>

Explicit Finite Difference Approximation Method for Solving the Riesz Space Fractional Percolation Equation

ABSTRACT This paper presents an efficient approximation method for solving the Riesz space fractional percolation equation mixed fractional derivative (RSFPEMFD) with the initial condition (IC) and derivative boundary conditions (DBC) using explicit finite difference method (EFDM). The shifted Grunwald estimate was used for analysis of the mixed fractional derivatives. In addition, the analysis method of consistency, stability, and convergence was used for analyzing the EFDM. To demonstrate the efficiency and validity of the proposed algorithm, some illustrative examples are given and comparing results with the exact solution. The results are displayed when necessary in tables using the MathCAD and MATLAB software package. The EFDM for solving RSFPEMFD has been demonstrated to be effective and reliable.

Регуляризованная модель типа фазового поля для описания системы «жидкость–твердое тело» с учетом химических реакций

We present regularized phase flield model for description of threephase «fluid–fluid–solid» system, where solid body obeys elastic rheology. It is assumed that one of the liquid phases chemically interacts with the solid one. As the corresponding equations of chemical kinetics, simple model equations describing a bimolecular reaction are considered. A feature of the model under consideration is its preliminary regularization according to quasi-hydrodynamic technique. A fully explicit finite difference approximation of this model is presented. Results of a simulation in two-dimensional setting are presented.

Development and validation of mathematical model of hydrotropic-reactive extraction of lignin

Abstract Hydrotropes have been largely explored as reactive extraction agent for lignin separation. In this paper, a mathematical model of hydrotropic-reactive extraction of sugarcane bagasse lignin was proposed and validated by experimental data from literature. The mathematical model was developed by assuming the particle is in slab shape, and by considering simultaneous processes of hydrotrope intra particle diffusion, second order reaction of lignin-hydrotrope, and intra-particle soluble delignification product diffusion. The proposed model results in a set of partial differential equations which were then solved by explicit finite difference approximation method. The mathematical model parameters were determined by fitting the model to the hydrotropic reactive extraction experimental data reported by Ansari and Gaikar (2014). Simulations show that the mathematical model of the hydrotropic-reactive extraction were well fitted to the experimental data with the obtained hydrotrope effective diffusivity (D eA ) of 5.0 × 10−11 m2/s, effective diffusivity of soluble lignin product (D eC ) of 9.0 × 10−12 m2/s and reaction rate constant (k r ) of 1.78 × 10−10 m3/(g.s). It was also observed that the reaction was first order to the hydrotrope (n = 1), and one half order to the lignin (m = 0.5). Meanwhile the pseudo-stoichiometric mass ratio of hydrotrope to lignin was 6.4 g hydrotrope/g lignin.

NUMERICAL SIMULATION OF THREE-DIMENSIONAL FLOW OF RADIATING GRAY NANOFLUID THROUGH POROUS MEDIUM SUBJECTED TO VIBRATIONAL ROTATIONS AND SLIP AT LIQUID-SHEET INTERFACE

We present a numerical scheme based on explicit finite difference approximation to handle the three-dimensional boundary layer flow of absorbing, emitting, and electrically conducting grey nanofluid over a flat surface. Nanofluid is designed by suspending iron-oxide nanoparticles (IONPs) in the base fluid. The flow field is assumed to be embedded in the porous medium, which executes vibrational rotations. To maintain the flow, velocity slip is introduced at the liquid-sheet interface. The governing equations are reduced to nondimensional form using dimensionless parameters and variables. The stability and convergence criteria have also been discussed to elaborate the validity of results. The hydromagnetic and convective heat transfer characteristics of magnetite nanofluid have been exhibited through graphs and tables for different related parameters. Substantial influences of oscillations and rotations have been noticed on the velocity profiles, temperature profiles, skin-friction coefficients, and Nusselt number.

Homotopy analysis method applied to second-order frequency mixing in nonlinear optical dielectric media

The classical problem of three-wave mixing in a nonlinear optical medium is investigated using the homotopy analysis method (HAM). We show that the power series basis builds a generic polynomial expression that can be used to study three-wave mixing for arbitrary input parameters. The phase-mismatched and perfectly phase matched cases are investigated. Parameters that result in generalized sum- and difference-frequency generation are studied using HAM with a power series basis and compared to an explicit finite-difference approximation. The convergence region is extended by increasing the auxiliary parameter.

Numerical Solution of Nonlinear Whitham-Broer-Kaup Shallow Water Model Using Finite Difference Methods

In this paper, we presented finite difference methods for solving nonlinear Whitham-Broer-Kaup (WBK) shallow water model numerically. We first subdivided the domain of the model by a net with a finite number of mesh points, and the derivative at each point replaced by explicit, Crank-Nicolson, and exponential finite difference approximations. The result is the system of algebraic equations which when solved, provide an approximation to the solutions of WBK model at the selected grid points. Also, a comparison has been made between the approximate solutions obtained by the proposed methods and the exact solutions. Numerical results represented in tables and figures with the help of MATLAB R2015a.

Time-space dependent fractional viscoelastic MHD fluid flow and heat transfer over accelerating plate with slip boundary

The MHD flow and heat transfer of viscoelastic fluid over an accelerating plate with slip boundary are investigated. Different from most classical works, a modified time-space dependent fractional Maxwell fluid model is proposed in depicting the constitutive relationship of the fluid. Numerical solutions are obtained by explicit finite difference approximation and exact solutions are also presented for the limiting cases in integral and series forms. Furthermore, the effects of parameters on the flow and heat transfer behavior are analyzed and discussed in detail.

A numerical method based on explicit finite difference for solving fractional hyperbolic PDE’s

In this paper, a new numerical scheme based on explicit finite difference approximation for solving fractional hyperbolic partial differential equations (FHPDE’s) is formulated. Numerical studies for the model problems are presented to confirm the accuracy and the effectiveness of the proposed method. The obtained results of proposed system are compared with exact solutions and the original system to show the efficient of the new method.

Temperature field simulation and phantom validation of a Two-armed Spiral Antenna for microwave thermotherapy.

In this paper, an Archimedean planar spiral antenna for the application of thermotherapy was designed. This type of antenna was chosen for its compact structure, flexible application and wide heating area. The temperature field generated by the use of this Two-armed Spiral Antenna in a muscle-equivalent phantom was simulated and subsequently validated by experimentation. First, the specific absorption rate (SAR) of the field was calculated using the Finite Element Method (FEM) by Ansoft's High Frequency Structure Simulation (HFSS). Then, the temperature elevation in the phantom was simulated by an explicit finite difference approximation of the bioheat equation (BHE). The temperature distribution was then validated by a phantom heating experiment. The results showed that this antenna had a good heating ability and a wide heating area. A comparison between the calculation and the measurement showed a fair agreement in the temperature elevation. The validated model could be applied for the analysis of electromagnetic-temperature distribution in phantoms during the process of antenna design or thermotherapy experimentation.

A toxin-mediated size-structured population model: Finite difference approximation and well-posedness.

The question of the effects of environmental toxins on ecological communities is of great interest from both environmental and conservational points of view. Mathematical models have been applied increasingly to predict the effects of toxins on a variety of ecological processes. Motivated by the fact that individuals with different sizes may have different sensitivities to toxins, we develop a toxin-mediated size-structured model which is given by a system of first order fully nonlinear partial differential equations (PDEs). It is very possible that this work represents the first derivation of a PDE model in the area of ecotoxicology. To solve the model, an explicit finite difference approximation to this PDE system is developed. Existence-uniqueness of the weak solution to the model is established and convergence of the finite difference approximation to this unique solution is proved. Numerical examples are provided by numerically solving the PDE model using the finite difference scheme.

Solving the Caputo Fractional Reaction-Diffusion Equation on GPU

We present a parallel GPU solution of the Caputo fractional reaction-diffusion equation in one spatial dimension with explicit finite difference approximation. The parallel solution, which is implemented with CUDA programming model, consists of three procedures: preprocessing, parallel solver, and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector-vector addition, and constant vector multiplication. The most time consuming loop of vector-vector addition and constant vector multiplication is optimized and impressive performance improvement is got. The experimental results show that the GPU solution compares well with the exact solution. The optimized GPU solution on NVIDIA Quadro FX 5800 is 2.26 times faster than the optimized parallel CPU solution on multicore Intel Xeon E5540 CPU.

Error Analysis of An Explicit Finite Difference Approximation for the Space Fractional Wave Equations

In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.

Fitting a Structured Juvenile–Adult Model for Green Tree Frogs to Population Estimates from Capture–Mark–Recapture Field Data

We derive point and interval estimates for an urban population of green tree frogs (Hyla cinerea) from capture-mark-recapture field data obtained during the years 2006-2009. We present an infinite-dimensional least-squares approach which compares a mathematical population model to the statistical population estimates obtained from the field data. The model is composed of nonlinear first-order hyperbolic equations describing the dynamics of the amphibian population where individuals are divided into juveniles (tadpoles) and adults (frogs). To solve the least-squares problem, an explicit finite difference approximation is developed. Convergence results for the computed parameters are presented. Parameter estimates for the vital rates of juveniles and adults are obtained, and standard deviations for these estimates are computed. Numerical results for the model sensitivity with respect to these parameters are given. Finally, the above-mentioned parameter estimates are used to illustrate the long-time behavior of the population under investigation.

Recharge assessment by means of vertical temperature profiles: analysis of possible influences

Temperature profiles (temperature as function of depth) can be used to derive vertical flow velocities or recharge rates. In many cases, solutions to the one-dimensional (1-D) heat transport equation are used, considering steady-state boundary conditions. Factors which can influence the derivation of the mean vertical flow velocity are studied here. Therefore, an explicit finite-difference approximation to the 1-D heat transport equation coupled with an inverse scheme was used to interpret temperature profiles. Measurement error (larger than 0.05°C) can result in important deviation of the derived mean flow velocity. Variation of vertical flow velocity as a function of time leads to asymmetric temperature envelopes. Yearly variation in vertical flow velocities, or temperature variations of the recharge water, also results in asymmetric temperature envelopes. Interpretation of these asymmetric envelopes leads to important differences between derived and actual mean vertical flow velocities. CitationVandenbohede, A. & Lebbe, L. (2010) Recharge assessment by means of vertical temperature profiles: analysis of possible influences. Hydrol. Sci. J. 55(5), 792–804.