The time-dependent fractional convection-diffusion (TFCD) equation is solved by the barycentric rational interpolation method (BRIM). Since the fractional derivative is the nonlocal operator, we develop a spectral method to solve the TFCD equation to get the coefficient matrix as a full matrix. First, the fractional derivative of the TFCD equation is changed to a nonsingular integral from the singular kernel to a density function. Second, efficient quadrature of the new Gauss formula are constructed to simply compute it. Third, matrix equation of discrete the TFCD equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, the convergence rate of BRIM is proved. Finally, a numerical example is given to illustrate our result.

A fractional cable (FC) equation is solved by the barycentric rational interpolation method (BRIM). As the fractional derivative is a nonlocal operator, we develop a spectral method to solve the FC equation to get the coefficient matrix as the full matrix. First, the fractional derivative of the FC equation is changed to a nonsingular integral from the singular kernel to the density function. Second, an efficient quadrature of a new Gauss formula is constructed to compute it simply. Third, a matrix equation of the discrete FC equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, convergence rate for FC equation of the BRIM is derived. At last, a numerical example is given to illustrate our results.

In this paper, we seek to solve the Kolmogorov-Petrovskii-Piskunov (KPP) equation by the linear barycentric rational interpolation method (LBRIM). As there are non-linear parts in the KPP equation, three kinds of linearization schemes, direct linearization, partial linearization, Newton linearization, are presented to change the KPP equation into linear equations. With the help of barycentric rational interpolation basis function, matrix equations of three kinds of linearization schemes are obtained from the discrete KPP equation. Convergence rate of LBRIM for solving the KPP equation is also proved. At last, two examples are given to prove the theoretical analysis.

This paper is concerned with solving a class of generalized Boussinesq shallow-water wave (GBSWW) equations by the linear barycentric rational collocation method (LBRCM), which are nonlinear partial differential equations (PDEs). By using the method of direct linearization, those nonlinear PDEs are transformed into linear PDEs which can be easily solved, and the corresponding differentiation matrix equations of their discretization linear GBSWW equations are also given by a Kronecker product. Based on the error estimate of a barycentric interpolation, the rates of convergence for numerical solutions of GBSWW equations are obtained. Finally, three examples are presented to show theoretical results.

Image encryption has been an important research topic in information security. Different from traditional encryption methods, heat flow cryptosystem is a new encryption method. This paper proposes an image encryption algorithm based on heat flow cryptosystem. First, a class of heat flow cryptosystem based on nonlinear pseudo-parabolic equations are given in this paper. Second, a numerical method with high precision namely barycentric Lagrange interpolation collocation method is proposed to solve the nonlinear pseudo-parabolic equation. Third, an image encryption algorithm based on the heat flow cryptosystem is designed, the detailed process of encryption and decryption algorithm is given, the flow diagram of algorithm is showed. Finally, the proposed encryption algorithm is applied to various image with gray and RGB format and compared with the current popular chaotic encryption algorithm. Many indicators such as histograms, information entropy and correlation are used to objectively evaluate the image encryption algorithm. The experimental results show that the proposed image encryption algorithm is better in most indicators and the algorithm is sensitive to the change of key and plaintext.

Barycentric rational collocation method (BRCM) for solving spatial fractional reaction-diffusion equation (SFRDE) is presented. New Gauss quadrature with weight function $(s_{\theta }-\tau {)}^{\xi -\alpha }$ is constructed to approximate fractional integral. Matrix equation of SFRDF is obtained from discrete SFRDE. With help of the error of barycentrix rational interpolation, convergence rate is obtained.

The barycentric rational collocation method for solving semi-infinite domain problems is presented. Following the barycentric interpolation method of rational polynomial and Chebyshev polynomial, matrix equation is obtained from discrete semi-infinite domain problem. Truncation method and transformation method are presented to solve linear and nonlinear differential equation defined on the semi-infinite domain problems. At last, three numerical examples are presented to valid our theoretical analysis.

Hypersingular integral equations have garnered extensive attention in the context of boundary element methods, particularly within natural boundary element methods. The asymptotic expansion of the composite rectangular rule's error function in Hadamard finite-part integrals yields a hypersingular kernel of $1/{\sin }^2(x-s)$. An extrapolation algorithm was developed to address this issue. To solve the hypersingular integral equation, we employed superconvergence points as collocation points, thereby constructing an extrapolation algorithm for hypersingular integral equations and establishing its convergence rate. A numerical example was provided to validate the efficacy of the method, corroborated by theoretical results that demonstrate the algorithm's effectiveness.

In this paper, composite trapezoidal quadrature for numerical evaluation of hypersingular integrals was first introduced. By Taylor expansion at the singular point $y$, error functional was obtained. We know that the divergence rate of $O(h^{-p}),p=1,2$, and there were no roots of the special function for the first part in the error functional. Meanwhile, for the second part of the error functional, the divergence rate was $O(h^{-p+1}),p=1,2$, but there were roots of the special function. We proved that the convergence rate could reach $O(h^2)$ at superconvergence points far from the end of the interval. Two modified trapezoidal quadratures are presented and their convergence rate can reach $O(h^2)$ at certain superconvergence points or any local coordinate point. At last, several examples were presented to test our theorem.