In this work, a new collocation approach using a combination of a wavelet operational matrix method and the exponential spline interpolation is proposed to solve the time-fractional convection-diffusion equation with variable coefficients. The operational matrix of fractional order integration is first derived based on sine-cosine wavelet functions, which helps to convert the underlying equation into a linear algebraic system. Then, an exponential B-spline method is introduced in spatial direction. On selecting a set of proper collocation points, the method in presence is evaluated on several test problems and the numerical results finally illustrate its validity and applicability.
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University（厦门大学数学科学学院、福建省数学建模与高性能科学计算重点实验室）
This paper presents a novel semi-analytical collocation method to solve multi-term variable-order time fractional partial differential equations (VOTFPDEs). In the proposed method it employs the Fourier series expansion for spatial discretization, which transforms the original multi-term VOTFPDEs into a sequence of multi-term variable-order time fractional ordinary differential equations (VOTFODEs). Then these VOTFODEs can be solved by using the recent-developed backward substitution method. Several numerical examples verify the accuracy and efficiency of the proposed numerical approach in the…
Hermite spectral collocation methods for fractional PDEs in unbounded domains
Southern University of Sciences and Technology（南方科技大学）；Hong Kong Baptist University（香港浸会大学）；Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）
In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree $p$, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, also in this case the given matrices are ill-conditioned both in the low and high frequencies for large $p$. More precisely, in the fractional scenario the symbol has a single zero at $0$ of order $α$, with $α$ the fraction…
An ADI Crank-Nicolson Orthogonal Spline Collocation Method for the Two-Dimensional Fractional Diffusion-Wave Equation
Mathematical Reviews, American Mathematical Society（美国数学学会数学评论）；College of Mathematics and Computer Science, Hunan Normal University（湖南师范大学数学与计算机科学学院）；School of Mathematics, Central South University（中南大学数学学院）
High order numerical methods based on quadratic spline collocation method and averaged L1 scheme for the variable-order time fractional mobile/immobile diffusion equation
Spectral method for substantial fractional differential equations
School of Mathematical Science, Xiamen University（厦门大学数学科学学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）；Beijing Computational Research Center（北京计算科学研究中心）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low order of convergence. For this reason, the development of classic numerical methods to solve such problems becomes a challenging issue. Since the non-smooth solutions have the same asymptotic behavior with polynomials of fractional powers, therefore, fractional basis functions are the best cand…
This paper develops a two-level fourth-order scheme for solving time-fractional convection-diffusion-reaction equation with variable coefficients subjected to suitable initial and boundary conditions. The basis properties of the new approach are investigated and both stability and error estimates of the proposed numerical scheme are deeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theory indicates that the method is unconditionally stable with convergence of order $O(k^{2-\fracλ{2}}+h^{4})$, where $k$ and $h$ are time step and mesh size, respectively, and $λ\in(0,1)$. This result sugge…
