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.

Approximation and application of the Riesz-Caputo fractional derivative of variable order with fixed memory

In this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.

A Quadratic Interpolation-Based Variational Bayesian Algorithm for Measurement Information Lost in Underwater Navigation

The main challenges of sequential estimations of underwater navigation applications are the internal/external measurement noise and the missing measurement situations. A quadratic interpolation-based variational Bayesian filter (QIVBF) is proposed to solve the underwater navigation problem of measurement information missing or insufficiency. The quadratic interpolation is used to improve the observed vector for the precision and stability of sequential estimations when the environment is changed or the measurement information is lost. The state vector, the predicted error covariance matrix, and the measurement noise matrix are derived based on the variational Bayesian method. Simulation results demonstrate the superiority of the proposed QIVBF compared with the traditional algorithm under the condition of measurement information lost by autonomous underwater vehicles.

Modified Differential Evolution Algorithm for Time-Modulated Linear Array Antenna

A modified differential evolution (MDE) algorithm based on a novel mutation strategy and adaptive adjustment strategy of parameter crossover rate (CR) is proposed to improve the population diversity and to avoid frapping in local optima. Also the simplified quadratic interpolation is employed to accelerate the convergence rate. Benchmark functions have been provided to verify the MDE algorithm. Compared with other improved evolutionary algorithms, experiment results reveal that the MDE has a promising performance in the convergence rate and the exploration ability. Finally, the proposed algorithm is proved to realize accelerating the optimization of time-modulated arrays (TMA).