On the Order of Convergence of Solutions of a Difference Equation to a Solution of the Diffusion Equation

1953 ◽  
Vol 1 (2) ◽  
pp. 111-135 ◽  
Author(s):  
M. L. Juncosa ◽  
D. M. Young
Keyword(s):  
Difference Equation ◽  
Diffusion Equation ◽  
Order Of Convergence ◽  
Convergence Of Solutions

scholarly journals Homogenization limit for the Diffusion Equation in a domain Perforated along (n - 1)-Dimensional Manifold with Dynamic Conditions on the Boundary of the Perforations: Critical Case

2019 ◽  
Vol 486 (1) ◽  
pp. 12-19
Author(s):  
M. N. Zubova ◽  
T. A. Shaposhnikova
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Critical Case ◽  
Original Problem ◽  
Dimensional Manifold ◽  
Dynamic Boundary Conditions ◽  
Dynamic Conditions ◽  
Convergence Of Solutions ◽  
Homogenization Model ◽  
With Memory

The problem of homogenization the diffusion equation in a domain perforated along an (n - 1)-dimensional manifold with dynamic boundary conditions on the boundary of the perforations is studied. A homogenization model is constructed that is a transmission problem for the diffusion equation with the transmission conditions containing a term with memory. A theorem on the convergence of solutions of the original problem to the solution of the homogenized one is proved.

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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