微分方程数值解 差分法解边值问题 陈文斌 multigridescicomput.com 复旦大学数学系 difference methods for BVP 2003, Fudan University, Chen Wenbin-p. 1/50
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TTest 50-25-15-10 difference methods for BVP 2003, Fudan University, Chen Wenbin-p. 2/50
Test 50-25-15-10 difference methods for BVP, 2003, Fudan University, Chen Wenbin – p.2/50
BVP 常微分方程两点边值问题 d 2, +qu=f (b)+(b) (0) 这里g,f∈C0(a,b),q≥0下面我们用差分法来求解这 个边值问题 difference methods for BVP 2003, Fudan University, Chen Wenbin-p. 3/50
BVP Lu ≡ −d2u dx2 + qu = f, x ∈ (a, b) u(a) = α0, u0(b) + βu(b) = α1 (0) q, f ∈ C0([a, b]), q ≥ 0 difference methods for BVP, 2003, Fudan University, Chen Wenbin – p.3/50
Difference Methods 区域的离散:将区间[a,分成N等分,分点为 a+ih,=0,1 h为步长,x;为节点 方程离散 边界条件的处理 difference methods for BVP 2003, Fudan University, Chen Wenbin-p. 4/50
Difference Methods [a, b] N x j = a + ih, i = 0, 1, . . . , N h x i difference methods for BVP, 2003, Fudan University, Chen Wenbin – p.4/50
Difference Methods 方程离散:由 Taylor公式,在节点x;处 (x)=n2((x+1)-20(x)+(x+1 (52) 这里余项(截断误差)为 R()= 则在x处将微分方程写成 (c2:-1)-2u(x:)+u+(x;)u(x)=f(x)+R difference methods for BVP 2003, Fudan University, Chen Wenbin-p. 5/50
Difference Methods Taylor x i u 00 ( x i) = 1 h 2 ( u ( x i+1 ) − 2 u ( x i) + u ( x i+1)) − h 2 12 u(4) ( ξ i ) R i ( u) = − h 2 12 u(4) ( ξ i ) x i − u ( x i + 1) − 2 u ( x i) + u ( x i − 1 ) h 2 + q ( x i ) u ( x i) = f ( x i)+ R i difference methods for BVP, 2003, Fudan University, Chen Wenbin – p.5/50