Symmetric collocation, which can be used to numerically solve linear partial differential equations, is a natural generalization of the well-established scattered data interpolation method known as radial basis function (rbf) interpolation. As with rbf interpolation, a major shortcoming of symmetric collocation is the high cost, in terms of floating point operations, of evaluating the obtained ...

The heat equation with Dirichlet boundary conditions is solved in various geometries by a modified Fourier collocation method. The computational domain is embedded in a larger, regular domain with a uniform, Cartesian grid, and the solution is defined to be identically zero outside the original domain. The discontinuities thus introduced across the boundary are handled by the modified Fourier c...

In this paper, we analyse some properties of the orthogonal collocation in the context of its use for reducing PDE (partial differential equations) chemical reactor models for numerical simulation and/or control design. The approximation of the first order derivatives is first considered and analysed with respect to the transfer of the stability properties of the transport component from the PD...

In this paper sixth order boundary value problems is solved numerically by collocation method. The solution is approximated as a linear combination of septic B-spline functions. The septic B-splines constitute a basis for the space of septic polynomial splines. In the method, the basis functions are redefined into a new set of basis functions which in number match with the number of selected co...

Tension spline is a function that, for given partition x0 < x1 < . . . < xn, on each interval [xi, xi+1] satisfies differential equation (D − ρiD)u = 0, where ρi’s are prescribed nonnegative real numbers. In the literature, tension splines are used in collocation methods applied to two-points singularly perturbed boundary value problems with Dirichlet boundary conditions. In this paper, we adap...

Heat conduction through eccentric systems with circular inner surface and regular polygonal outer surface has been analyzed. Constant wall temperature at the inner surface and convective heat transfer from the outer surface is considered. A two dimensional semi analytical method using boundary collocation technique is used in which collocation points are considered on the outer periphery of the...

A more careful count of the operations involved in solving the linear system associated with collocation of a two-point boundary value problem using rough splines reverses results recently reported by others in this journal. In addition, it is observed that the use of the technique of "condensation of parameters" can decrease the computer storage required. Furthermore, the use of a particular h...

The polynomial chaos (PC) method has been widely adopted as a computationally feasible approach for uncertainty quantification (UQ). Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a nonlinear system of equations at every time step. Using the Galerkin approach the size of the system state increase...

We present a Python module named PyCheb, to solve the ordinary differential equations by using spectral collocation method. PyCheb incorporates discretization using Chebyshev points, barycentric interpolation and iterate methods. With this Python module, users can initialize the ODEsolver class by passing attributes, including the both sides of a given differential equation, boundary conditions...

Combining the variational iteration method (VIM) with the sparse grid theory, a dynamic sparse grid approach for nonlinear PDEs is proposed in this paper. In this method, a multilevel interpolation operator is constructed based on the sparse grids theory firstly. The operator is based on the linear combination of the basic functions and independent of them. Second, by means of the precise integ...