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

Abstract

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.

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

“…Proof. Equations 15 and 17 can be found in Qiu et al 22 and Ciarlet, 45 and (18) is referred to Borggaard et al 46 We only need to prove (16). In fact, for any , ∈ R 2 , using Lagrange mean value theorem, we have…”

“…The numerical analysis of the Stokes or Navier‐Stokes equations has been researched extensively (see previous studies). Some studies also have been devoted to the Stokes equations with damping.…”

“…1-3 The damping comes from the resistance to the motion of the flow, which describes various physical situations such as porous media flow, drag, or friction effects and some dissipative mechanisms. 4,5 The numerical analysis of the Stokes or Navier-Stokes equations has been researched extensively (see previous studies [6][7][8][9][10][11][12][13][14][15][16][17][18] ). Some studies also have been devoted to the Stokes equations with damping.…”

Numerical solutions of the radiosity equation for a spherical quatrefoil on Mars

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“…The operator ( ) exists and is bounded on ) (U C and L 2 (U) (Arqub et al, 2017). The dimension for the approximating subspace of spherical polynomials of degree N  is…”