Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition

Abstract

The time-fractional heat conduction equation with the Caputo derivative and with heat absorption term proportional to temperature is considered in a sphere in the case of central symmetry. The fundamental solution to the Dirichlet boundary value problem is found, and the solution to the problem under constant boundary value of temperature is studied. The integral transform technique is used. The solutions are obtained in terms of series containing the Mittag-Leffler functions being the generalization of the exponential function. The numerical results are illustrated graphically.

Studying heat conduction in a sphere considering hybrid fractional derivative operator

“…In this paper, we consider the following time-fractional heat conduction equation with heat absorption term in spherical coordinates in the case of central symmetry [6] ( )…”

“…Recently, fractional calculus is used to study many real world problems formulated in the form of fractional partial differential equations [1][2][3][4][5][6]. Many definitions for the fractional derivative are proposed in the literature [7][8][9][10][11][12].…”

“…Many definitions for the fractional derivative are proposed in the literature [7][8][9][10][11][12]. Some examples of these definitions are Caputo, Riemann-Liouville, He's fractional derivative, generalized fractional derivatives and Reisz definitions [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Investigating heat conduction in a sphere with heat absorption using generalized Caputo fractional derivative

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] To find analytical solutions to fractional differential equations, some appropriate transforms are used for achieving this task, such as ρ-Laplace transform and finite sin-Fourier transform. [16][17][18][19] In this paper, we use ρ-Laplace transform and finite sin-Fourier transform to solve the timefractional heat equation associated with heat absorption in spherical coordinates with central symmetry, which is given by 19…”

“…Equation ( 1) with the conditions (3)-( 5) was investigated in [19] when ρ = 1 and β = 0. In this paper, we get the closed-form solutions of Equation ( 1) with the conditions (3)-( 5) when ρ > 0 and β > −1.…”