Model And Numerical Simulation Of Fractional Heat Transfer Differential Equations

Heat transfer is a widespread physical phenomenon both in nature and life. The classical theory of heat transfer is based on the integer order differential and integral calculus. However, it is difficult to accurately describe the heat transfer law of viscous material and biological tissue by using classical integer order equation, because of the time memory and path dependence existence in the heat transfer.Firstly, the development history and research status of fractional calculus was introduced in this paper, the definitions and properties of common fractional differential operator were briefly summarized as well. The author reviews the analysis process of the classical heat conduction model, and the application of fractional order calculus in heat transfer has been introduced. Secondly, theoretically the one-dimensional fractional heat conduction model of rigid body has been studied. In Cattaneo equation, the integer derivative operator has been replaced by fractional order derivative operator, meanwhile the local heat flux has been plugged in the transform equation to overcome the limited conditions of fractional order operator. Finally, generalized Fourier’s law and the orthogonal curvilinear coordinates system have been implicated in time fractional heat conduction equation, spherical coordinate time fractional heat conduction equation was obtained. The paper shows how the heat of fractional order transfers in solid sphere, separated variable method is used to get analytic solution in the form of triple series. The initial boundary value conditions have been restricted, taking a series of limited items to get the numerical solution.Based on the analysis of the heat transfer equation of fractional order theory and the related model, this paper provides a new mathematical tool for the study of the complex heat transfer phenomena, in particularly when it comes to the heat transfer problem which has time memory character, fractional order model is superior to the integer order model. Figure11; Table1; Reference 64...