In this paper, we present a numerical scheme based on collocation method to solve stochastic non-linear Poisson–Boltzmann equations (PBE). This equation is generalized version of the arising from form biomolecular modeling case. Applying radial basis functions (RBFs) allows us deal with difficulties complexity domain. To indicate accuracy RBF method, results for two-dimensional models, also stu...

We generalize existing Jacobi–Gauss–Lobatto collocation methods for variable-order fractional differential equations using a singular approximation basis in terms of weighted Jacobi polynomials of the form (1 ± x)μP a,b j (x), where μ > −1. In order to derive the differentiation matrices of the variable-order fractional derivatives, we develop a three-term recurrence relation for both integrals...

In solving semilinear initial boundary value problems with prescribed non-periodic boundary conditions using implicit-explicit and implicit time stepping schemes, both the function and derivatives of the function may need to be computed accurately at each time step. To determine the best Chebyshev collocation method to do this, the accuracy of the real space Chebyshev differentiation, spectral ...

We evaluate the performance of global stochastic collocation methods for solving nonlinear parabolic and elliptic problems (e.g., transient and steady nonlinear di↵usion) with random coe cients. The robustness of these and other strategies based on a spectral decomposition of stochastic state variables depends on the regularity of the system’s response in outcome space. The latter is a↵ected by...

In this paper, we propose and analyze a spectral approximation for the numerical solutions of fractional integro-differential equations with weakly kernels. First, original are transformed into an equivalent singular Volterra integral equation, which possesses nonsmooth solutions. To eliminate singularity solution, introduce some suitable smoothing transformations, then use Jacobi collocation m...

In this paper, we present a mixed spline/spectral method to solve Volterra delay-integro-differential equations (VDIDEs). This method is based on generating the sextic spline collocation methods in all subintervals. The approximation of the integration in these subintervals, can be calculated by the El-Gendi method. Convergence results of the present method are presented. Numerical results are ...

In this paper we discuss the issue of conservation and convergence to weak solutions of several global schemes, including the commonly used compact schemes and spectral collocation schemes, for solving hyperbolic conservation laws. It is shown that such schemes, if convergent boundedly ahnost everywhere, will converge to weak solutions. The results are extensions of the classical Lax-Wendroff t...

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of FredholmVolterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L∞ norm and weighted L2-norm. The numerical examp...