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Comparative Analysis of Caputo and Variable-Order Fractional Derivative algorithms Across Various Applications

International Journal of Applied and Computational Mathematics 2025年第3期11卷 1-40页

作者： Amilo, David Sadri, Khadijeh Hincal, Evren Mathematics Research Center Near East University Nicosia 99138 Turkey Department of Mathematics Near East University Nicosia 99138 Turkey Faculty of Art and Science University of Kyrenia Kyrenia Turkey

Fractional derivative models have emerged as powerful tools for describing complex dynamics in various fields. However, selecting the appropriate fractional derivative method for specific applications remains a challenge due to the diverse nature of real-world systems. This paper presents a comprehensive comparison between the Caputo fractional derivative and the variable-order fractional derivative methods for multiple applications. By exploring diverse examples, we illustrate the strengths and limitations of each approach in modeling real-world phenomena. numerical simulations are conducted using the Grunwald–Letnikov method to approximate the solutions. In addition, we propose a new algorithm for comparing and implementing fixed and variable-order fractional derivatives efficiently. The motivation for this study stems from the growing demand for accurate, adaptable, and computationally efficient tools to model systems with memory and dynamic changes. The results reveal distinct differences in performance and accuracy, highlighting the scenarios in which each method excels. This comparative study aims to provide information for researchers and practitioners in selecting the appropriate fractional derivative model and algorithm for their specific applications. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2025.

关键词： Caputo derivative operator Mathematical model numerical algorithm Variable-order derivative operator

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The Artificial Neural Network Optimization for Thermally Magnetized Williamson Fluid Flow over a Porous Surface

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International Journal of Applied and Computational Mathematics 2025年第2期11卷 1-20页

作者： Priyadharshini, P. Karpagam, V. Department of Mathematics PSG College of Arts and Science Tamil Nadu Coimbatore 641014 India

The current scientific work examines a pseudo-plastic Williamson fluid over a homogenous porous media. Momentum and energy equations refer to the executive branch partial differential equations (PDE). A new transformation and symmetry analysis scaling group are applied to these PDEs to convert them into ordinary differential equations (ODEs). The numerical algorithm utilized to perform the BVP4C solver from the MATLAB series solves these ODEs. The present research explores the impact of different physical parameters on temperature and velocity distributions. The coefficient of skin friction (Cfx) and the local Nusselt number (Nux) are employed to illustrate graphs and table values. Furthermore, Machine Learning (ML) models adopted Multiple Linear Regression (MLR) to predict physical quantities with a 95% accuracy. Subsequently, implementing the Levenberg-Marquardt Artificial Neural Network approach handles complex nonlinear problems, accurate predictions, optimizations, and error reduction to validate the outcomes of biomedical applications and its accuracy level of 99%. These innovative approaches are expected to promote future developments in energy-related technology optimization, and the conclusions are more advanced than previous physical evidence and biological applications, including blood flow analysis, drug delivery systems, and cancer treatment. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2025.

关键词： Artificial neural network Multiple linear regression numerical algorithm Porous surface Symmetry analysis

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Computation of numerical Pade-Hermite and simultaneous Pade systems .2. A weakly stable algorithm

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1996年第2期17卷 268-297页

作者： Cabay, S Jones, AR Labahn, G BELL NO RES LTD OTTAWAON K1Y 4H7CANADA UNIV WATERLOO DEPT COMP SCIWATERLOOON N2L 3G1CANADA

For k + 1 power series a(0)(z),...,a(k)(z), we present a new iterative, look-ahead algorithm for numerically computing Pade-Hermite systems and simultaneous Pade systems along a diagonal of the associated Pade tables. The algorithm computes the systems at all those points along the diagonal at which the associated striped Sylvester and mosaic Sylvester matrices are well conditioned. The operation and the stability of the algorithm is controlled by a single parameter tau which serves as a threshold in deciding if the Sylvester matrices at a point are sufficiently well conditioned. We show that the algorithm is weakly stable and provide bounds for the error in the computed solutions as a function of tau. Experimental results are given which show that the bounds reflect the actual behavior of the error. The algorithm requires O(\\n\\(2) + s(3)\\n\\) Operations to compute Pade-Hermite and simultaneous Fade systems of type n = [n(0),...,n(k)], where \\n\\ = n(0)+...+n(k) and s is the largest step-size taken along the diagonal. An additional application of the algorithm is the stable inversion of striped and mosaic Sylvester matrices.

关键词： Pade-Hermite approximants simultaneous Pade approximants striped Sylvester inverses mosaic Sylvester inverses numerical algorithm numerical stability

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On the Calculation of the Boiler Thermal Diagram 2nd

On the Calculation of the Boiler Thermal Diagram

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2nd International Conference on Information Technologies and Their Applications

作者： Zhurbenko, Mykola Bardadym, Tamara NASU VM Glushkov Inst Cybernet Kyiv Ukraine

ISBN: (纸本)9783031734168;9783031734175

The complex technical object usually can be described as a set of technological components (blocks). Its functional diagram can be represented by an oriented multi-graph. The vertices of the graph correspond to separate components (blocks) of a technical object. The blocks are characterized by design parameters. The graph may contain feedback loops. Blocks' output parameters can be determined from the values of their input parameters. The software developed in C++ language serves for calculating functional diagrams of steam boilers. The software determines also the values of derived output parameters from the design parameters values. It can be used to improve the technical characteristics of an object and determine control parameters for various modes of its operation. The calculation of the thermal scheme is based on the use of a model heat exchanger. In this case, the calculation of the thermal scheme is reduced to an iterative solution of a system of linear equations. A brief description of the software is provided.

关键词： Steams Boilers Functional Diagram Optimal Design numerical algorithm Software

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APPROXIMATION METHODS FOR HYBRID DIFFUSION SYSTEMS WITH STATE-DEPENDENT SWITCHING PROCESSES: numerical algorithmS AND EXISTENCE AND UNIQUENESS OF SOLUTIONS

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 2010年第6期41卷 2335-2352页

作者： Yin, G. Mao, Xuerong Yuan, Chenggui Cao, Dingzhou Wayne State Univ Dept Math Detroit MI 48202 USA Univ Strathclyde Dept Stat & Modelling Sci Glasgow G1 1XH Lanark Scotland Univ Wales Swansea Dept Math Swansea SA2 8PP W Glam Wales

Focusing on hybrid diffusions in which continuous dynamics and discrete events coexist, this work is concerned with approximation of solutions for hybrid stochastic differential equations with a state-dependent switching process. Iterative algorithms are developed. The continuous-state-dependent switching process presents added difficulties in analyzing the numerical procedures. Weak convergence of the algorithms is established by a martingale problem formulation first. This weak convergence result is then used as a bridge to obtain strong convergence. In this process, the existence and uniqueness of the solution of the switching diffusions with continuous-state-dependent switching are obtained. In contrast to existing results of solutions of stochastic differential equations in which the Picard iterations are utilized, Euler's numerical schemes are considered here. Moreover, decreasing-stepsize algorithms together with their weak convergence are given. numerical experiments are also provided for demonstration.

关键词： switching diffusion numerical algorithm convergence existence and uniqueness of strong solution

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A new algorithm for numerical modelling of impurity transport in the frame of statistically homogeneous sharply contrasting media

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RUSSIAN JOURNAL OF numerical ANALYSIS AND MATHEMATICAL MODELLING 2019年第6期34卷 339-351页

作者： Kondratenko, Petr S. Matveev, Leonid V. Vasiliev, Alexander D. Russian Acad Sci Nucl Safety Inst Bolshaya Tulskaya St 52 Moscow 115191 Russia State Univ Moscow Inst Phys & Technol 9 Inst Per Dolgoprudnyi 141700 Moscow Region Russia

A new method is developed to calculate characteristics of contaminant transport (including non-classical regimes) in statistically homogeneous sharply contrasting media. A transport integro-differential equation in the space-time representation is formulated on the basis of the model earlier proposed by one of the authors (L. M.). Analytical expressions for transport characteristics in limiting time intervals in the one-dimensional case are derived. An interpolation form is proposed for the integral kernel of the transport equation. On a basis of this expression, an algorithm is developed for numerical modelling the contaminant transport in statistically homogeneous sharply contrasting media. Trial numerical 1D calculations are performed based on this algorithm. Good agreement was found between the numerical simulation results and the asymptotic analytical expressions.

关键词： Non-equilibrium transport model non-classical transport quasi-diffusion sub-diffusion concentration asymptotics numerical algorithm

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ELIAS: An Accurate and Extensible Lithography Aerial Image Simulator With Improved numerical algorithms

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IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING 2009年第2期22卷 276-289页

作者： Yu, Peng Pan, David Z. Univ Texas Austin Dept Elect & Comp Engn Austin TX 78712 USA

Lithography simulators have been playing an indispensable role in process optimization and design for manufacturability (DFM). The ever smaller feature sizes demand higher numerical accuracy and faster runtime on these lithography simulators. Aerial image simulation is the first key step in lithography simulation, and the method using transmission cross coefficient (TCC), which is a two-dimensional integral, is the most commonly used technique for full-chip aerial image simulation. In this paper, we present a very accurate, yet efficient and extensible aerial image simulator, ELIAS. We find that the majority of the numerical error during the TCC computation is due to the discontinuous boundaries of the support of the TCC integrand. We reduce the error dramatically by using a recursive integration algorithm. Because TCC is usually computed on uniform grids, we further speed up the algorithm without increasing the errors. Given the same accuracy, our new algorithm can speed up the runtime by 100 x -1000 X. Our algorithm also provides smooth tradeoff between accuracy and runtime. It can be used to benchmark other lithography aerial simulators. In addition, ELIAS provides an open-source, flexible software framework to incorporate different lithography settings.

关键词： Accuracy aerial image simulation C plus ELIAS fast Fourier transform (FFT) Hopkins equation lithography simulation numerical algorithm recursive integration runtime transmission cross coefficient (TCC)

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On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations

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COMPUTATIONAL & APPLIED MATHEMATICS 2018年第4期37卷 4966-4976页

作者： Assanova, A. T. Kadirbayeva, Zh. M. Inst Math & Math Modeling Dept Differential Equat 125 Pushkin Str Alma Ata 050010 Kazakhstan

A linear two-point boundary value problem for a system of loaded differential equations with impulse effect is investigated. Values in the previous impulse points are taken into consideration in the conditions of impulse effect. The considered problem is reduced to an equivalent multi-point boundary value problem for the system of ordinary differential equations with parameters. A numerical implementation of parametrization method is offered using the Runge-Kutta method of 4th-order accuracy for solving the Cauchy problems for ordinary differential equations. The constructed numerical algorithms are illustrated by examples.

关键词： Loaded differential equation Impulse effect Parametrization method numerical algorithm

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Fragmentation of numerical algorithms for parallel subroutines library

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JOURNAL OF SUPERCOMPUTING 2011年第2期57卷 161-171页

作者： Kireev, S. Malyshkin, V. Russian Acad Sci Inst Computat Math & Math Geophys Siberian Branch Novosibirsk Russia

Fragmentation is a well-known method of the parallelization of numerical algorithms and programs. algorithm fragmentation allows creating fragmented parallel programs that can be executed on parallel computers of different types (multiprocessors and/or multicomputers) and can be dynamically tuned to all the available resources. Fragmentation of the often used numerical algorithms, their representation for inclusion into the library of parallel numerical subroutines and properties of the runtime system are considered.

关键词： Asynchronous programming Parallel program numerical algorithm Fragment-based programming Dynamic programs' properties

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numerical solution of the Schrodinger equations via a reliable algorithm

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INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND MATHEMATICS 2015年第5期6卷 417-424页

作者： Heidari, Mohammad Borzabadi, Akbar Hashemi Damghan Univ Sch Math & Comp Sci Damghan Iran

In this paper, a reliable algorithm for solving Schrodinger equations is established. By second-order central difference scheme, the second-order spatial partial derivative of the Schrodinger equations are reduced to a system of first-order ordinary differential equations, that are solved by an efficient algorithm. The comparison of the numerical solution and the exact solution for some test cases shows that the given algorithm is easy and practical for extracting good approximate solutions of Schrodinger equations.

关键词： Schrodinger equation finite difference method numerical algorithm approximation

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