This paper explores the development of students! knowledge of mathematical procedures. Students! tendency to develop rote knowledge of procedures has been widely commented on. An alternative, more flexible endpoint for the development of procedural knowledge is explored here, where students choose to deviate from established solving patterns on particular problems for greater efficiency. Studen...

In this article we study the convergence of a stochastic particle system that interacts through threshold hitting times towards a novel equation of McKeanVlasov type. The particle system is motivated by an original model for the behavior of a network of neurons, in which a classical noisy integrate-and-fire model is coupled with a cable equation to describe the dendritic structure of each neuron.

In this paper, by using the theories and methods of integral equation and computer algebra, a reliable algorithm for solving the Volterra integral equation is established, and a new Maple algorithm mainproc is established, too. Some examples are presented to illustrate the implementations of the algorithm. The results of the examples indicate that the algorithm of Taylor polynomial method is si...

Traditionally, solving the adjoint equation for unsteady problems involves solving a large, structured linear system. This paper presents a variation on this technique and uses a Monte Carlo linear solver. The Monte Carlo solver yields a forward-time algorithm for solving unsteady adjoint equations. When applied to computing the adjoint associated with Burgers’ equation, the Monte Carlo approac...

Lane-Emden equation is a nonlinear singular equation that plays an important role in the astrophysics. In this paper, we have applied the collocation method based on rational Chebyshev functions to solve Lane-Emden type equations. The method reduces solving the nonlinear ordinary differential equation to solving a system of nonlinear algebraic equations. The comparison of the results with the o...

In this paper we prove that the Diophantine equation x − 4cxy + (6c+ 2)xy + 4cxy + y = 1, where c ≥ 3 is an integer, has only the trivial solutions (±1, 0), (0,±1). Using the method of Tzanakis, we show that solving this quartic Thue equation reduces to solving the system of Pellian equations (2c+ 1)U − 2cV 2 = 1, (c− 2)U − cZ = −2, and we prove that all solutions of this system are given by (U...