The computation of the exponential of a tridiagonal matrix and its applications have always been of interest. One application considered here is when the method of lines is used to solve the heat equation, where the equation is transformed into a system of ordinary differential equations (ODEs), and this system has a solution that depends on the exponential of a tridiagonal Toeplitz matrix. Strang and MacNamara presented an approximate method for computing the exponential of a symmetric tridiagonal Toeplitz matrix that appears in the solution of ODEs. Their method is based on approximating the entries of the exponential matrix with modified Bessel functions of the first kind at certain values, and accordingly, the exponential matrix is decomposed as the difference of a Toeplitz matrix and a Hankel matrix. Here, we aim to extend this idea to the general case of tridiagonal Toeplitz matrices and stabilize the method by approximating the matrix exponential with a banded matrix, which makes the complexity of computing the exponential matrix independent of the matrix size. Additionally, we provide an error analysis for these methods and a bound for the entries of the exponential of the tridiagonal Toeplitz matrices. As a main contribution of this work, the idea is implemented to solve the heat equation, and the uniform stability of the method is proved. By using a splitting approach, the method is generalized for two-dimensional problems. Numerical illustrations demonstrate the efficiency of the new methods and bounds.
Application of the Parallel Dichotomy Algorithm for solving Toeplitz tridiagonal systems of linear equations with one right-hand side
Institute of Computational Mathematics and Mathematical Geophysics（计算数学与数学地球物理研究所）；Budker Institute of Nuclear Physics（布德克核物理研究所）；Novosibirsk State University（新西伯利亚国立大学）
On $τ$-preconditioners for a quasi-compact difference scheme to Riesz fractional diffusion equations with variable coefficients
Hanshan Normal University（韩山师范学院）；Southwestern University of Finance and Economics（西南财经大学）；University of Insubria（因苏布里亚大学）；University of Uppsala（乌普萨拉大学）
Explicit unconditionally stable methods for the heat equation via potential theory
Flatiron Institute, Simons Foundation（西蒙斯基金会平顿研究所）；Department of Mathematics, University of Pennsylvania（宾夕法尼亚大学数学系）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Department of Mathematics Sciences, New Jersey Institute of Technology（新泽西理工学院数学科学系）
