Solutions of a Class of the Heat Conduction and Membrane Vibration Problems with Time Fractional Derivative

Solutions of a class of the membrane vibration and heat conduction problems with time fractional derivative and sine-waving on the boundary were studied. Firstly, the Taylor series is employed to expand independent variables of the boundary, so that the small parameter only affect the boundary but independent variables. Then multiple scales were introduced into the original equation and the boundary, the equations on the 0 power of small parameter and the 1 power of small parameter were developed. The approximate solution of the equations of the membrane vibration problem and the heat conduction problem about the 0 power of small parameter were obtained. With the definition and properties of Riemann-Liouville fractional derivative and fractional integration, the change rules of the solutions of the membrane vibration problem and the heat conduction problem were discussed, respectively. The effects from fractional derivative of the membrane vibration problem and the heat conduction problem on the original problem were developed.