Solving fractional differential equations by the ultraspherical integration method

In this paper, we present a numerical method to solve a linear fractional differential equations. This new investigation is based on ultraspherical integration matrix to approximate the highest order derivatives to the lower order derivatives. By this approximation the problem is reduced to a constrained optimization problem which can be solved by using the penalty quadratic interpolation method. Numerical examples are included to confirm the efficiency and accuracy of the proposed method.

An Immersed Boundary Method for Calculating the Relative Viscosity of a Suspension of Rigid Particles

In this paper, we provide a numerical framework to calculate the relative viscosity of a suspension of rigid particles. A high-resolution background grid is used to solve the flow around the particles. In order to generate infinite number of particles in the suspension, a particle is placed in the center of a cubic cell and periodic boundary conditions are imposed in two directions. The flow around the particle is solved using the second-order accurate curvilinear immersed boundary (CURVIB) method [1]. The particle is discretized with triangular elements, and is treated as a sharp interface immersed boundary by reconstructing the velocities on the fluid nodes adjacent to interface using a quadratic interpolation method. Hydrodynamic torque on the particle has been calculated, to solve the equation of motion for the particle and obtain its angular velocity. Finally, relative viscosity of the suspension has been calculated based on two different methods: (1) the rate of the energy consumption and (2) bulk stress-bulk strain method. The framework has been validated by simulating a suspension of spheres, and comparing the numerical results with the corresponding analytical ones. Very good agreement has been observed between the analytical and the calculated relative viscosities using both methods. This framework is then used to model a suspension with arbitrary complex particles, which demonstrates the effect of shape on the effective viscosity.

Study on Control Algorithm of Wind-Solar Hybrid Generation System

An improved maximum power point tracking method for wind-solar hybrid generation system was proposed in this paper. The maximum power point was tracked by Perturbation and Observation Algorithms and quadratic interpolation method. Taking the advantage of the above methods, the new method could avoid shock phenomenon of Perturbation and Observation Algorithms near the maximum power point, reduce miscarriage of justice. The whole system was simulated on MATLAB. The simulation results shown that, the improved control method could search the maximum power point of wind-solar hybrid generation system quickly, and work stably.