Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions

This work deals with the pseudospectral method to solve the Sturm–Liouville eigenvalue problems with Caputo fractional derivative using Chebyshev cardinal functions. The method is based on reducing the problem to a weakly singular Volterra integro-differential equation. Then, using the matrices obtained from the representation of the fractional integration operator and derivative operator based on Chebyshev cardinal functions, the equation becomes an algebraic system. To get the eigenvalues, we find the roots of the characteristics polynomial of the coefficients matrix. We have proved the convergence of the proposed method. To illustrate the ability and accuracy of the method, some numerical examples are presented.

On the Approximate Solution of Partial Integro-Differential Equations Using the Pseudospectral Method Based on Chebyshev Cardinal Functions

In this paper, we apply the pseudospectral method based on the Chebyshev cardinal function to solve the parabolic partial integro-differential equations (PIDEs). Since these equations play a key role in mathematics, physics, and engineering, finding an appropriate solution is important. We use an efficient method to solve PIDEs, especially for the integral part. Unlike when using Chebyshev functions, when using Chebyshev cardinal functions it is no longer necessary to integrate to find expansion coefficients of a given function. This reduces the computation. The convergence analysis is investigated and some numerical examples guarantee our theoretical results. We compare the presented method with others. The results confirm the efficiency and accuracy of the method.

Новый численный метод решения нелинейных стохастических интегральных уравнений

The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by other works and it is efficient to use for different problems.

A hybrid method for solving time fractional advection–diffusion equation on unbounded space domain

In this article, a hybrid method is developed for solving the time fractional advection–diffusion equation on an unbounded space domain. More precisely, the Chebyshev cardinal functions are used to approximate the solution of the problem over a bounded time domain, and the modified Legendre functions are utilized to approximate the solution on an unbounded space domain with vanishing boundary conditions. The presented method converts solving this equation into solving a system of algebraic equations by employing the fractional derivative matrix of the Chebyshev cardinal functions and the classical derivative matrix of the modified Legendre functions together with the collocation technique. The accuracy of the presented hybrid approach is investigated on some test problems.

Cardinal Functions of Purely Atomic Measures

Let $$\mu $$ μ be a purely atomic finite measure. Without loss of generality we may assume that $$\mu $$ μ is defined on $${\mathbb {N}}$$ N , and the atoms with smaller indexes have larger masses, that is $$\mu (\{k\})\ge \mu (\{k+1\})$$ μ ( { k } ) ≥ μ ( { k + 1 } ) for $$k\in {\mathbb {N}}$$ k ∈ N . By $$f_\mu :[0,\infty )\rightarrow \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ f μ : [ 0 , ∞ ) → { 0 , 1 , 2 , ⋯ , ω , c } we denote its cardinal function $$f_{\mu }(t)=\vert \{A\subset {\mathbb {N}}:\mu (A)=t\}\vert $$ f μ ( t ) = | { A ⊂ N : μ ( A ) = t } | . We study the problem for which sets $$R\subset \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ R ⊂ { 0 , 1 , 2 , ⋯ , ω , c } there is a measure $$\mu $$ μ such that $$R=\text {rng}(f_\mu )$$ R = rng ( f μ ) . We are also interested in the set-theoretic and topological properties of the set of $$\mu $$ μ -values which are obtained uniquely.