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.
Highlights
The classical parabolic heat conduction equation with the source term proportional to temperature∂T = a T − bT ∂t (1)Communicated by José Tenreiro Machado.Y
In the last few decades, differential equations with derivatives of non-integer order attract the attention of the researchers as such equations provide a very suitable tool for description of many important phenomena in physics, geophysics, chemistry, biology, engineering and solid mechanics
This transform is the convenient reformulation of the sin-Fourier series and is used for Dirichlet boundary condition with the prescribed boundary value of a function, since for the Laplace operator in the case of central symmetry we have d2 f (r ) dr 2
Summary
In the last few decades, differential equations with derivatives of non-integer order attract the attention of the researchers as such equations provide a very suitable tool for description of many important phenomena in physics, geophysics, chemistry, biology, engineering and solid mechanics Equation (2) results from the time-nonlocal generalization of the Fourier law with the “long-tail” power kernel. Such a generalization can be interpreted in terms of derivatives and integrals of non-integer order. 2, we find the fundamental solution to the Dirichlet boundary value problem using the Laplace transform with respect to time t and the finite sin-Fourier transform with respect to the spatial coordinate r.
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