Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of ene...

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Bibliographic Details
Published in:International journal of numerical methods for heat & fluid flow Vol. 28; no. 4; pp. 828 - 856
Main Author: Abu Arqub, Omar
Format: Journal Article
Language:English
Published: Bradford Emerald Publishing Limited 03.04.2018
Emerald Group Publishing Limited
Subjects:
Algorithms
Applied mathematics
Boundary conditions
Circuit design
Circuits
Computation
Computational fluid dynamics
Computer simulation
Conduction heating
Conductive heat transfer
Convergence
Differential equations
Differential thermal analysis
Dynamics
Energy conservation
Engineering
Exact solutions
Fluid dynamics
Fluid flow
Groundwater
Groundwater treatment
Heat conduction
Heat transfer
Hilbert space
Hydrodynamics
Hydrologic models
Hydrology
Kernel functions
Mathematical analysis
Mathematical functions
Mathematical models
Multiphase flow
Nonlinear equations
Numerical methods
Numerical models
Partial differential equations
Physics
Realism
Representations
Series (mathematics)
Solutions
Solvers
Heat
Gas dynamics
Reproducing kernel algorithm
Time-fractional partial differential equations
Navier–Stokes
Burgers’ and Fitzhugh–Nagumo equations
Robin boundary conditions
Advection–diffusion
ISSN:0961-5539, 1758-6585
Online Access:Get full text
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Summary:Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.
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ISSN:0961-5539
1758-6585
DOI:10.1108/HFF-07-2016-0278