Artificial Neural Networks: Methods and Applications in Fractional Order Systems

This is the first book that uses Artificial Neural Networks (ANN) to solve fractional order systems. As a powerful data modeling tool, information is processed through neurons in parallel manner to solve a specific problem. Knowledge is acquired through learning and stored with inter neuron connections strength which are expressed by numerical values called weights. These weights are used to complete output signal values for new testing input signal value.In this book, multi-layer ANN model will be used to handle fractional order differential equations (FDEs). The network is trained using a back-propagation unsupervised learning algorithm which is based on the gradient descent rule. The ANN approximate solution of FDEs may be expressed as a sum of two terms; the first part satisfies boundary or initial conditions, and the second term contains ANN output with network parameters (weights and biases).Next, single layer Functional Link Artificial Neural Network (FLANN) models will be included for solving the FDEs. In FLANN the hidden layer is replaced by a functional expansion block for enhancement of the input patterns using orthogonal polynomials such as Chebyshev, Legendre, Hermite, etc. The computations become efficient because the procedure does not need to have hidden layer. Thus, the numbers of network parameters are less than the traditional ANN model.Varieties of FDEs will be addressed to show the reliability and efffectiveness of ANN. Singular nonlinear fractional Lane-Emden type equations, fractional vibration problems viz. Bagley-Torvik equations, fractional electrical problems viz. RLC, RC, LC circuit problems, Duffing oscillator problems with fractional derivatives etc. will be handled using multi-layer ANN and single layer FLANN models.