Solving Diffusion Equation Using a New Multiquadric Quasi-interpolation

In this paper, a new univariate quasi- interpolation operator is presented by means of construction way with cubic Multiquadric functions. It possesses univariate cubic polynomial reproduction property, quasi convexity- preserving and shapepreserving of order 4 properties, and a higher convergence rate. First, the quasi- interpolation operator L-R(x) is applied to approximate the derivative of order m=1,2,3. and its approximation capacity is obtained, i. e., C(0)h(h+c)(3-m)+C(1)c(2). Second, it is used to construct numerical schemes to solve the diffusion equation. Using the derivative of the quasi- interpolation to approximate the spatial derivative of the differential equation. And applying Crank- Nicolson scheme and back Euler scheme to approximate the temporal derivative of the differential equation. And as c=O(h(2)), the computational accuracy of the scheme is both O(Delta t(2)+h(2)) and O(Delta t(2)+h(2)) respectively. Finally, some numerical examples is given to verify the scheme for the onedimensional diffusion equation. The numerical results show that the numerical solution are very close to the exact solution.