Pedro M. Lima

Mario M. Graça, Pedro M. Lima

We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with $n+1$ nodes is used the resulting iterative method has convergence order at least $n+2$, starting with the case $n=0$ (which corresponds to the Newton's method).

Pedro M. Lima, Evelyn Buckwar

We introduce a new numerical algorithm for solving the stochastic neural field equation (NFE) with delays. Using this algorithm we have obtained some numerical results which illustrate the effect of noise in the dynamical behaviour of stationary solutions of the NFE, in the presence of spatially heterogeneous external inputs.

Pedro M. Lima, Evelyn Buckwar

In the present paper we are concerned with a numerical algorithm for the approximation of the two-dimensional neural field equation with delay. We consider three numerical examples that have been analysed before by other authors and are directly connected with real world applications. The main purposes are 1) to test the performance of the mentioned algorithm, by comparing the numerical results with those obtained by other authors; 2) to analyse with more detail the properties of the solutions and take conclusions about their physical meaning.

Somayeh Nemati, Pedro M. Lima, Delfim F. M. Torres

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann-Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann-Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.

Pedro M. Lima, Evelyn Buckwar

We are concerned with the numerical solution of a class integro-differential equations, known as Neural Field Equations, which describe the large-scale dynamics of spatially structured networks of neurons. These equations have many applications in Neuroscience and Robotics. We describe a numerical method for the approximation of solutions in the two-dimensional case, including a space-dependent delay in the integrand function. Compared with known algorithms for this type of equation we propose a scheme with higher accuracy in the time discretisation. Since computational efficiency is a key issue in this type of calculations, we use a new method for reducing the complexity of the algorithm. The convergence issues are discussed in detail and a number of numerical examples is presented, which illustrate the performance of the method.