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Abstract. A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. Consider the heat equation ∂u ∂t = γ ∂2u ∂x2, 0 < x < ℓ, t ≥ 0, (11.8) representing a bar of length ℓ and constant thermal diﬀusivity γ > 0. To be concrete, we impose time-dependent Dirichlet boundary conditions u(t,0) = α(t), u(t,ℓ) = β(t), t ≥ 0, (11.9) specifying the temperature at the ends of the bar, along with the initial conditions u(0,x) = f(x), 0 ≤ x ≤ ℓ, (11.10)

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The fourth method is a finite volume method on cartesian grids to simulate compressible Euler or Navier Stokes Flows in complex domains. An immersed boundary-like technique is developed to take into account boundary conditions around the obstacles with order two accuracy. Finite Difference Methods ... 2.12 Neumann boundary conditions 29 ... 9.4 Stiffness of the heat equation 186 9.5 Convergence 189

Broad overview of "discretization" of unknown function (finite difference, finite elements, spectral methods, boundary elements), "discretization" of PDE (finite differences, Galerkin, integral equations), imposition of boundary conditions, and solution of resulting linear (or nonlinear system) by iterative methods. Poisson equation (14.3) is to be solved on the square domain subject to Neumann boundary condition To generate a finite difference approximation of this problem we use the same grid as before and Poisson equation (14.3) is approximated at internal grid points by the five-point stencil. The time fractional heat conduction in an infinite plate of finite thickness, when both faces are subjected to boundary conditions of second kind, has been studied. The time fractional heat conduction equation is used, when attempting to describe transport process with long memory, where the rate of heat conduction is inconsistent with the classical Brownian motion. The stability and ...