Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

Abstract

Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

Formation of Intermetallic Phases in Ion Implantation

“…e implicit-difference scheme has been suggested for the solution of diffusion kinetic problem describing ion implantation by intermetallic phase formation. For further interesting models and methods, we refer the readers to [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. We actually suggest a model of the surface modification of nickel-aluminum ions with the relaxation of mass flows.…”

On semilocal convergence of three-step Kurchatov method under weak condition

“…4, the computational efficiencies indexes of method (7) have larger values than methods (2) and (4) for m ≥ 9. Here red lines used for method (2), green lines used for method (4) and blue lines used for method (7).…”

“…Authors also found the better result for order of convergence, optimal computational efficiency and efficiency index of method (7) as comparison from methods (2) and (4). One of the important benefits of this method is the evaluation of only one divided difference of the operator in each step, whereas order of convergence of the method is 3.…”

“…Ahmad et al [2] found the numerical solution of three-term time-fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. Kumar et al [30] worked on fourth-order derivativefree iterative methods for finding a multiple root.…”

Accurate spectral algorithm for two‐dimensional variable‐order fractional percolation equations

“…Fractional calculus 24‐32 is a division of calculus theory, such that the differential equations are superb relevant to represent many phenomena in different fields like mechanics, biology, 26,31 chemistry, 25,27‐30 viscoelasticity, 32 engineering, finance, and physics 33 fields. Samko and Ross 34 and Lorenzo and Hartley 35 were introduced the variable‐order fractional concept for first time.…”