Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order

In this paper, we develop a Hermite cubic spline collocation method (HCSCM) for solving variable-order nonlinear fractional differential equations, which apply C1-continuous nodal basis functions to an approximate problem. We also verify that the order of convergence of the HCSCM is about O(hmin{4−α,p}) while the interpolating function belongs to Cp(p≥1), where h is the mesh size and α the order of the fractional derivative. Many numerical tests are performed to confirm the effectiveness of the HCSCM for fractional differential equations, which include Helmholtz equations and the fractional Burgers equation of constant-order and variable-order with Riemann-Liouville, Caputo and Patie-Simon sense as well as two-sided cases.

Pricing American bond options using a cubic spline collocation method

In this paper, American options on a discount bond are priced under the Cox-Ingrosll-Ross (CIR) model. The linear complementarity problem of the option value is solved numerically by a penalty method. The problem is transformed into a nonlinear partial differential equation (PDE) by adding a power penalty term. The solution of the penalized problem converges to the one of the original problem. To numerically solve this nonlinear PDE, we use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of trapezoidal method and a cubic spline collocation method, respectively. We show that this full discretization scheme is second order convergent, and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. Numerical results are presented and compared with other collocation methods given in the literature.

Cubic Spline Collocation Method for Fractional Differential Equations

We discuss the cubic spline collocation method with two parameters for solving the initial value problems (IVPs) of fractional differential equations (FDEs). Some results of the local truncation error, the convergence, and the stability of this method for IVPs of FDEs are obtained. Some numerical examples verify our theoretical results.