Algorithm solution for space-fractional diffusion equations

In this study, we propose approximate algorithm solution of the space-fractional diffusion equation (SFDE's) based on a quarter-sweep (QS) implicit finite difference approximation equation. To derive this approximation equation, the Caputo's space-fractional derivative has been used to discretize the proposed problems. By using the Caputo's finite difference approximation equation, a linear system will be generated and solved iteratively. In addition to that, formulation and implementation algorithm the Quarter-Sweep AOR (QSAOR) iterative method are also presented. Based on numerical results of the proposed iterative method, it can be concluded that the proposed iterative method is superior to the FSAOR and HSAOR iterative method

Numerical Solution for Solving Space-Fractional Diffusion Equations using Half-sweep Gauss-seidel Iterative Method

The main purpose of this paper is to examine theeffectiveness of Half-Sweep Gauss-Seidel (HSGS) method forSpace-Fractional diffusion equations. The Caputo’s derivativeand implicit finite difference scheme will be used to discretizelinear space-fractional equation of the first order to constructsystem linear equation. The basic formulation and application ofthe HSGS iterative method are also presented. Two numericalexamples and comparison with other iterative methods showsthat the present method is effective. Based on computationalnumerical result, the solution obtained by proposed iterativemethod is in excellent agreement, it can be concluded that theproposed iterative method is superior to the Full-Sweep Gauss-Seidel (FSGS) iterative metho

Komputasi Numerik Metode Iteratif Half-Sweep Preconditioned Gauss-Seidel Untuk Memecahkan Persamaan Resepan Pecahan Waktu

Dalam penelitian ini, peneliti berusaha memperoleh persamaan aproksimasi beda hingga dari diskritisasi persamaan resapan pecahan waktu linier satu dimensi dengan menggunakan turunan pecahan waktu Caputo. Suatu sistem persamaan linier akan dibuat dengan menggunakan persamaan aproksimasi beda hingga Caputo. Kemudian hasil dari system persamaan linier tersebut diselesaikan dengan menggunakan metode iterasif numerik Half-Sweep Preconditioned Gauss-Seidel (HSPGS) dimana efektivitasnya akan dibandingkan dengan metode Preconditioned Gauss-Seidel (PGS), (dikenal juga sebagai Full-Sweep Preconditioned Gauss- Seidel (FSPGS)) dan Gauss-Seidel (GS) sebagai metode kontrol. Contoh masalah juga disajikan untuk menguji efektivitas metode yang diusulkan. Temuan penelitian ini menunjukkan bahwa metode iteratif yang diusulkan yaitu HSPGS lebih unggul dibandingkan dengan metode FSPGS dan GS

Numerical Solution of a Kind of Fractional Parabolic Equations via Two Difference Schemes

A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. Therefore, we presented the numerical solution via the explicit and the implicit schemes. We presented together the stability and convergence of this time-fractional parabolic equation with two difference schemes. The explicit and the implicit schemes in this case are stable under some conditions

Parallel Direct and Iterative Methods for Solving the Time-Fractional Diffusion Equation on Multicore Processors

The work is devoted to developing the parallel algorithms for solving the initial boundary problem for the time-fractional diffusion equation. After applying the finite-difference scheme to approximate the basis equation, the problem is reduced to solving a system of linear algebraic equations for each subsequent time level. The developed parallel algorithms are based on the Thomas algorithm, parallel sweep algorithm, and accelerated over-relaxation method for solving this system. Stability of the approximation scheme is established. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to compare these methods and to study the performance of parallel implementations. The parallel sweep method shows the lowest computing time. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.Funding: The first author (M.A.S.) and fourth author (E.N.) were financially supported by the Ministry of Education and Science of the Republic of Kazakhstan (project AP09258836). The second author (E.N.A.) and third author (V.E.M.) received no external funding

Modelagem de processos usando equações diferenciais parciais fracionárias

Orientador : Prof. Dr. Marcelo Kaminski LenziCoorientador: Prof. Dr. David Alexander MitchelDissertação (mestrado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Engenharia Química. Defesa: Curitiba, 30/04/2010Bibliografia: fls. 56-60Resumo: O emprego de técnicas de modelagem de processos químicos é de fundamental importância para descrição matemática dos mesmos e a aplicação de técnicas de controle de processo que venham garantir sua operação segura e competitiva. Neste trabalho, aplicou-se o ferramental baseado em equações diferenciais de ordem fracionárias para a modelagem de sistemas de engenharia química. Mais especificamente, foram estudadas e aplicadas técnicas numéricas para problemas não reportados na literatura, concernentes à sistemas com geometria radial. Em uma segunda etapa, foram analisadas a mistura de sólidos e a dispersão axial sob a ótica de equações diferenciais fracionárias. A partir de dados experimentais previamente reportados na literatura, foram estimados parâmetros de modelos representados por equações diferenciais de ordem fracionária tipo parcial. Considerando a técnica heurística de algoritmos genéticos, foram estimados parâmetros do modelo de ordem fracionária e de ordem inteira para comparação. Em ambos os estudos, mistura de sólidos e dispersão axial, o modelo fracionário levou à menores valores da função objetivo usada para estimação de parâmetros. Para mistura de sólidos o modelo fracionário obteve FOBJ = 0,0480 e o modelo inteiro obteve FOBJ = 0,0501. Para dispersão axial, o modelo fracionário obteve FOBJ = 0,0593 e o modelo inteiro obteve FOBJ = 0,0766. Desta forma, o ajuste dos pontos experimentais mostrou-se melhor pelo modelo fracionário, o que pode ser comprovado pela inspeção visual dos gráficos comparativos, o que comprava a viabilidade do uso de equações diferenciais fracionárias para a modelagem de sistemas de engenharia química.Abstract: The use of process modeling techniques plays a key role for mathematical description of chemical processes and the consequent use of process control techniques which allow a safer and competitive operation. In this work, fractional differential equations were used to model chemical engineering systems. More specifically, numerical techniques were studied applied to solve equations not reported in the literature, mainly concerning radial systems. In a second step, solid mixture and axial dispersion were considered for modeling purposes using fractional differential equations. From experimental data previously reported in the literature, parameters were estimated in order to obtain a fractional partial differential equation based model to adequately describe the data. The heuristic technique of genetic algorithms was considered for parameter estimation and as benchmark of comparison integer order models were also obtained. In both studies, i.e., solid mixing and axial dispersion, the fractional based model lead to lower values of the objective function used for parameter estimation. For solid mixing studies, the fractional model lead to FOBJ = 0,0480, while the integer model lead to FOBJ = 0,0501. For axial dispersion, the fractional model lead to FOBJ = 0,0593 and the integer model lead to FOBJ = 0,0766. Consequently, the fractional model provided a better experimental data fit, which can also be proved by visual inspection of comparative plots. Therefore, fractional differential equations can be successfully used for chemical engineering systems modeling

New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

Análise da variação térmica sazonal em barragem de contrafortes com uso de cálculo fracionário

Orientador : Profª. Drª. Liliana Madalena GramaniCo-orientador : Prof. Dr. Eloy KaviskiTese (doutorado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa: Curitiba, 04/05/2016Inclui referências : f. 114-119Área de concentração : Programação matemáticaResumo: Barragens de concreto respondem a diversos tipos de carregamentos, sejam eles provenientes da força da gravidade, pressão hidrostática da água ou variação térmica sazonal. Dentre estes, diversos trabalhos presentes na literatura destacam a importância de se avaliar os efeitos da temperatura no desempenho estrutural de barragens. Altos índices de variação de temperatura podem afetar o desempenho, resistência e durabilidade das estruturas, de modo que a correta avaliação do campo de temperaturas é essencial para que uma futura análise e determinação das tensões de origem térmica possa ser realizada. Através de um estudo de caso na Barragem da Usina Hidrelétrica de Itaipu (UHI), é proposto um esquema numérico via elementos finitos para calibrar e validar os parâmetros térmicos do concreto (condutividade térmica, calor e massa específica) de um bloco de contraforte quando este está sujeito à variação térmica sazonal. Devido ao advento das aplicações do Cálculo Fracionário (que se trata de uma abordagem generalizada do Cálculo Tradicional, onde derivadas e integrais sucessivas podem ser de ordem não inteira) esta tese traz como inovação a modelagem térmica em barragem de contrafortes através de Equações Diferenciais Fracionárias. As soluções de um modelo via Cálculo Tradicional e via Cálculo Fracionário foram analisadas comparativamente aos dados observados por meio do MAPE (Erro Percentual Absoluto Médio). Para o caso da barragem da UHI, com geometria, condições de contorno e propriedades do concreto bem definidas, o modelo térmico fracionário apresentou um resultado mais próximo à realidade do bloco em estudo ao verificar que o maior MAPE nos pontos de prova foi de 3.23%. O objetivo assim é instigar futuras aplicações do Cálculo Fracionário no âmbito de barragens. Palavras-chaves: Modelagem Térmica. ANSYS. Equação do Calor Fracionária. Métodos Aproximados.Abstract: Concrete dams respond to various types of loads, whether from the force of gravity, hydrostatic water pressure or seasonal thermal variation. Among these, several studies in the literature highlight the importance of evaluating the effects of temperature on the structural performance of dams. High temperature change rates may affect the performance, strength and durability of the structures, so that the correct evaluation of the temperature field is essential for future analysis and determination of the thermal stress can be performed. Through a case study in Itaipu Hydroelectric Power Plant dam (UHI), a numerical scheme is proposed via finite element to calibrate and validate the thermal parameters of concrete (thermal conductivity, heat and density) of a buttress block when this it is subject to seasonal thermal variation. Due to the advent of applications of Fractional Calculus (which is a generalized approach to the traditional calculation, where derivatives and successive integrals can be no integer order) this thesis brings as innovation thermal modeling in buttresses dam through Fractional Differential Equations. The solutions of a model via traditional calculation and via Fractional Calculus were analyzed in comparison to the observed data through MAPE (Mean Absolute Percent Error). In the case of UHI dam, with geometry, boundary conditions and concrete properties well defined, the fractional thermal model presented closer to reality block in study results to check the biggest MAPE in test points was 3.23%. The purpose is to instigate so future applications of fractional calculus under dams. Key-words: Thermal Modeling. ANSYS. Fractional Heat Equation. Approximate method

Simulación de sistemas reacción-transporte mediante formulaciones de ecuaciones integrales

Este trabajo se centra en el uso de las formulaciones de ecuaciones integrales (FEI) para el desarrollo de esquemas numéricos para sistemas reacción-transporte. Las FEI consisten en transformar los operadores diferenciales a operadores integrales, permitiendo la incorporación exacta de las condiciones a la frontera y la reducción en la propagación del error ocasionada por la discretización de derivadas numéricas. Además, los esquemas integrales se caracterizan por su metodología sistemática y su estructura matemática que permite la fácil interpretación física de los procesos involucrados en los sistemas reacción-transporte. Por otro lado, es bien sabido que los esquemas basados en la discretización de los operadores diferenciales como son las diferencias finitas no estándar (DFNE), proveen mejores aproximación que los esquemas clásicos de diferencias finitas. Sin embargo, dichos esquemas se obtienen a partir de expansiones de Taylor truncadas y de reglas Heurísticas. Con el objeto de sistematizar la metodología de obtención de los esquemas de DFNE, en trabajos recientes se ha demostrado que dichos esquemas pueden ser obtenidos como un caso particular de las FEI. En el desarrollo de los esquemas de diferencias finitas basadas en formulaciones integrales (DFFI) no es necesario el uso de expansiones de series de Taylor o reglas Heurísticas; únicamente se emplean reglas de cuadratura para las integrales. Además, el esquema DFFI incorpora factores de ponderación en la discretizaci´on de los nodos en la frontera que mejoran la aproximación numérica del esquema tradicional diferencias finitas. En este trabajo se extienden los resultados de DFFI a problemas dinámicos tipo reaccióntransporte con condiciones a la frontera generales, bajo ciertas condiciones reproducen los resultados reportados en la literatura. Más aún, se suponen diferentes esquemas de discretización de las integrales en las FEI, lo que conduce a esquemas DFFI con estructura no local (i.e., que consideran nodos adyacentes al punto evaluado en el término fuente). Debido a que los esquemas de DFFI se basan en metodologías sistemáticas no es difícil extenderlos a procesos reaccióndifusión-convección y/o a procesos de mayor dimensionalidad (e.g., 2 dimensiones). Finalmente, se muestra la generalidad de los esquemas integrales extendiendo los resultados encontrados a problemas reacción-transporte descritos por derivadas de orden arbitrario. Particularmente, se desarrollan los esquemas numéricos para la solución sistemas reacción-difusión fraccionales