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標題: 利用B-spline 計算 Monge-Amp`ere方程式
Numerical Methods for the Monge-Amp`ere Equation by B-spline Functions
作者: 朱冠華
Kuan-Hua Chu
關鍵字: B-spline;Monge-Amp`ere方程式;B-spline;Monge-Amp`ere Equation
引用: C. de Boor, A practical guide to splines, Springer-Verlag (1978). N. Caglar, AND H. Caglar, B-spline solution of singular boundary value problems, Appl. Math. Comput. 182 (2006), no. 2, 1509--1513. H. N. Caglar, S. H. Caglar, AND E.H. Twizell, The numerical solution of fifth-order boundary-value problems with sixth-degree B-spline functions, Appl Math Lett 12 (1999), pp. 25--30. H. N. Caglar, S. H. Caglar, AND E. H. Twizell, The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions, Int. J. Comput. Math. 71 (1999), no. 3, pp. 373--381. P. R. Halmos, Finite Dimensional Vector Spaces, Springer-Verlag (1974). K. Hollig, Finite element methods with B-splines, Society Industrial and Applied Mathematics (2003). M. K. Kadalbajoo, AND K. C. Patidar, Exponentially fitted spline in compression for the numerical solution of singular perturbation problems, Comput. Math. Appl. 46 (2003), no. 5--6, 751--767. M. K. Kadalbajoo, AND P. Arora, B-spline collocation method for the singular-perturbation problem using artificial viscosity, Comput. Math. Appl. 57 (2009), no. 4, 650-663. K. R. Koch, AND M. Schmidt, N-dimensional B-spline surface estimated by lofting for locally improving IRI, 1(1), (2011), 41--45. P. Massopust, Interpolation and Approximation with Splines and Fractals, Oxford University Press (2010). G. Nurnberger, Approximation by Spline Functions, Springer-Verlag (1989). L. A. Piegl, The NURBS Book, Springer-Verlag, 2nd edition (1997). B. F. Plybon, An Introduction to Applied Numerical Analysis, Pws Pub Co (1992). P. M. Prenter, Splines and Variational Methods, Wiley (1989). M. Schatzman, Numerical Analysis: A Mathematical Introduction, Springer-Oxford University Press (2002). G. Strang, Introduction to linear algebra, Wellesley-Cambridge Press (2009). K. N. S. K. Viswanadham, P. M. Krishna, AND C. P. Rao, Numerical Solution of Fifth Order Boundary Value Problems by Collocation Method with Sixth Order B-Splines, Int. J. Appl. Sci. Eng., (2010). 8, 2: 119-125.
In this thesis, we used the collection method with B-splines as basis functions for the numerical solutions of the partial differential equation
au_{xx} (x,y) + bu_{xy}(x,y) +cu_{yy}(x,y) =f(x,y), in $Omega$.
u(x,y)=g(x,y), on partialOmega.
If a=c and b=0 then the equation in (0.1) is the well-known Poisson equation.
In order to consider the approximations of the
Monge-Amp`ere equation, the purpose of this research is considering approximationsof solutions of (0.1).
For instant, let lambda_1(u) and lambda_2(u) be the eigenvalues of D^2u where u satisfies det D^2 u = f(u) in Omega and u= g on partial Omega.
Then u satisfies
riangle u = sqrt{lambda_1^2(u) + lambda_2^2(u) +2 f(u) }.

Thus, one can carry out the following iterations:
1. Choose the first guess u^0.
2. Solve u^1 from riangle u^1 = sqrt{2f(u^0)}.
3. Solve u^{n+1} from riangle u^{n+1} = sqrt{2f(u^n) + lambda _1^2(u^n) + lambda^2_2(u^n);}.

We consider the Dirichlet problem in dimension two.
In this research, a new basis is provided which
reduces the number of basis functions and coincides with that of the knots.
The reducing process is using the boundary conditions to redefine the basis.
The approximation solutions are solved by the argument of the tensor product of matrices.
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