*Article* **Time-Fractional Heat Conduction in a Plane with Two External Half-Infinite Line Slits under Heat Flux Loading**

### **Yuriy Povstenko 1,\* and Tamara Kyrylych 2**


Received: 22 April 2019; Accepted: 15 May 2019; Published: 18 May 2019

**Abstract:** The time-fractional heat conduction equation follows from the law of conservation of energy and the corresponding time-nonlocal extension of the Fourier law with the "long-tail" power kernel. The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with two external half-infinite slits with the prescribed heat flux across their surfaces. The integral transform technique is used. The solution is obtained in the form of integrals with integrand being the Mittag–Leffler function. A graphical representation of numerical results is given.

**Keywords:** fractional calculus; Caputo derivative; generalized Fourier law; Laplace transform; Fourier transform; Mittag–Leffler function
