High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Jianxiong Cao ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Dirichlet Boundary ◽  
Order Approximation ◽  
High Order ◽  
Dirichlet Boundary Conditions ◽  
High Order Accuracy ◽  
High Order Approximation ◽  
Order Accuracy ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
New Formula

AbstractIn this paper, we first establish a high-order numerical algorithm for α-th (0 < α < 1) order Caputo derivative of a given function f(t), where the convergence rate is (4 − α)-th order. Then by using this new formula, an improved difference scheme with high order accuracy in time to solve Caputo-type fractional advection-diffusion equation with Dirichlet boundary conditions is constructed. Finally, numerical examples are carried out to confirm the efficiency of the constructed algorithm.

scholarly journals A New High-Order Approximation for the Solution of Two-Space-Dimensional Quasilinear Hyperbolic Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
R. K. Mohanty ◽  
Suruchi Singh
Keyword(s):  
Dirichlet Boundary ◽  
Hyperbolic Equations ◽  
Order Approximation ◽  
High Order ◽  
Dirichlet Boundary Conditions ◽  
Polar Coordinates ◽  
Hyperbolic Partial Differential Equation ◽  
High Order Approximation ◽  
Linear Hyperbolic Equation ◽  
Derivation Procedure

we propose a new high-order approximation for the solution of two-space-dimensional quasilinear hyperbolic partial differential equation of the form , , , subject to appropriate initial and Dirichlet boundary conditions , where and are mesh sizes in time and space directions, respectively. We use only five evaluations of the function as compared to seven evaluations of the same function discussed by (Mohanty et al., 1996 and 2001). We describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Some examples and their numerical results are provided to justify the usefulness of the proposed method.

scholarly journals Symplectic multiquadric quasi-interpolation approximations of KdV equation

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5161-5171
Author(s):  
Shengliang Zhang ◽  
Liping Zhang
Keyword(s):  
Kdv Equation ◽  
Order Approximation ◽  
High Order ◽  
Basis Functions ◽  
High Order Accuracy ◽  
Approximation Properties ◽  
High Order Approximation ◽  
Shape Preserving ◽  
Order Accuracy ◽  
Long Time

Radial basis functions quasi-interpolation is very useful tool for the numerical solution of differential equations, since it possesses shape-preserving and high-order approximation properties. Based on multiquadric quasi-interpolations, this study suggests a meshless symplectic procedure for KdV equation. The method has a number of advantages over existing approaches including no need to solve a resultant full matrix, accuracy and ease of implementation. We also present a theoretical framework to show the conservativeness and convergence of the proposed method. As the numerical experiments show, it not only offers a high order accuracy but also has a good property of long-time tracking capability.

High-order approximation for generalized fractional derivative and its application

2019 ◽  
Vol 29 (9) ◽  
pp. 3515-3534 ◽  
Author(s):  
Swati Yadav ◽  
Rajesh K. Pandey ◽  
Anil K. Shukla ◽  
Kamlesh Kumar
Keyword(s):  
Diffusion Equation ◽  
Taylor Expansion ◽  
Order Of Convergence ◽  
Order Approximation ◽  
High Order ◽  
Generalized Derivative ◽  
Content Type ◽  
High Order Scheme ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

Purpose This paper aims to present a high-order scheme to approximate generalized derivative of Caputo type for μ ∈ (0,1). The scheme is used to find the numerical solution of generalized fractional advection-diffusion equation define in terms of the generalized derivative. Design/methodology/approach The Taylor expansion and the finite difference method are used for achieving the high order of convergence which is numerically demonstrated. The stability of the scheme is proved with the help of Von Neumann analysis. Findings Generalization of fractional derivatives using scale function and weight function is useful in modeling of many complex phenomena occurring in particle transportation. The numerical scheme provided in this paper enlarges the possibility of solving such problems. Originality/value The Taylor expansion has not been used before for the approximation of generalized derivative. The order of convergence obtained in solving generalized fractional advection-diffusion equation using the proposed scheme is higher than that of the schemes introduced earlier.

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