Implicit finite difference approximation for time fractional heat conduction under boundary condition of second kind

Abstract

The time fractional heat conduction in an infinite plate of finite thickness, when both faces are subjected to boundary conditions of second kind, has been studied. The time fractional heat conduction equation is used, when attempting to describe transport process with long memory, where the rate of heat conduction is inconsistent with the classical Brownian motion. The stability and convergence of this numerical scheme has been discussed and observed that the solution is unconditionally stable. The whole analysis is presented in dimensionless form. A numerical example of particular interest has been studied and discussed in details.

The Numerical Approximation of the Bioheat Equation of Space-Fractional Type Using Shifted Fractional Legendre Polynomials

“…Multiplying by ) ) ) generates algebraic equations for ) ) ) Then, by integrating from to and using the orthogonal property, we have ) ) ) with the initial condition from equation (2) in the matrix form is as follows:…”

“…In many cases, finding the exact solutions for these equations is difficult or impossible. Therefore, researchers used approximate or numerical solution methods [1][2][3][4][5]. The fractional calculus is utilized to improve the modeling accuracy of many phenomena in natural sciences.…”

A New Mixed Nonpolynomial Spline Method for the Numerical Solutions of Time Fractional Bioheat Equation

“…The problem of the time fractional Pennes bioheat transfer equation for the modeling of skin tissue heat transfer is expressed in previous works [6,14,15,16] [2][3][4][5][6], [8][9][10][11][12][13][14], [16], [17][18][19][20][21] as: ∫…”

“…Definition 2: The Caputo fractional derivative of order is defined by [2][3][4][5][6], [8][9][10][11][12][13][14], [16], [19][20][21] as: ∫…”

“…Eq. (13) can lead to ∑ (14) By multiplying (13) and (14) by and respectively, then adding these equations, we get ( ) ∑ ( ) ( ) (15) From (11) and by substituting the value in (15), the last equation can be rewritten in the following form ( ) ∑ ( ) ( ) (16) Equation 16contains linear algebraic equations by unknowns . Thus, we need two boundary equations when and and by using Taylor series for these nodes, we get the following two equations:…”