Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions

Abstract

A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential) accuracy of the proposed method.

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam ∗

“…en, these coefficients are substituted into (14), and the approximate solution is obtained. For more details, see [36].…”

On some wavelet solutions of singular differential equations arising in the modeling of chemical and biochemical phenomena

“…Jacobi wavelet is the family of wavelets reduced into Legendre wavelet, Chebyshev wavelet, and Gegenbauer wavelet for the specific value of κ and ω. There are a lot of research papers available for the solution of ordinary and partial differential equations using Jacobi and Bernoulli wavelets, for instance, see [1,10,20,46]. In this study, we introduce two methods based on Jacobi and Bernoulli wavelets for solving models of electrohydrodynamic flow in a circular cylindrical conduit, nonlinear heat conduction model in the human head, spherical catalyst equation, and spherical biocatalyst equation.…”

Solution of third-order Emden–Fowler-type equations using wavelet methods

“…Raza and Khan (2019) have dealt with the numerical solution of neutral delay differential equation (NDDE) by applying Haar wavelet. Faheem et al (2020) investigated NDDE via Gegenbauer and Bernoulli wavelets, and Zogheib et al (2017) used computational method based on Bernoulli wavelet for solving diffusion and wave equation. Rahimkhani and Ordokhani (2019), on the other hand, solved fractional partial differential equation using Bernoulli wavelet.…”