121 lines
4.2 KiB
Go
121 lines
4.2 KiB
Go
// ODEDiff
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/*
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------------------------------------------------------
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作者 : Black Ghost
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日期 : 2018-12-26
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版本 : 0.0.0
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------------------------------------------------------
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差分方法求解常微分方程
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理论：
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对于常微分方程：x''(t) = p(t)x'(t)+q(t)(t)+r(t)
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机器边值x(a) = x0, x(b) = xN
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使用中心差分公式可得
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x_(j+1)-2xj+x_(j-1) x_(j+1)-x_(j-1)
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--------------------- = pj----------------- + qj*xj+rj
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h^2 2h
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即
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-h h
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(---pj-1)x_(j-1) + (2+h^2*qj)xj + (---pj-1)x_(j+1) = -h^2*rj
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2 2
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下标j表示*(tj), tj=a+j*h，(区间[a, b]等分为N等份)
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整理成N-1阶线性方程组:
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|2+h^2*q1 h*p1/2-1 |
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|-h*p2/2-1 2+h^2*q2 h*p2/2-1 |
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| -h*p3/2-1 2+h^2*q3 h*p3/2-1 |*
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| ...... |
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| -h*p_(N-2)/2-1 2+h^2*q_(N-2) h*p_(N-2)/2-1|
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| -h*p_(N-1)/2-1 2+h^2*q_(N-1)|
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[x1 x2 x3 ... x_(N-1)]' =
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[-h^2*r1+e0 -h^2*r2 -h^2*r3 ... -h^2*r_(N-2) -h^2*r_(N-1)+eN]'
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其中：
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e0 = (h*p1/2+1)x0, eN = (h*p_(N-1)/2+1)xN
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参考：John H. Mathews and Kurtis D. Fink. Numerical
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methods using MATLAB, 4th ed. Pearson
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Education, 2004. ss 9.9
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------------------------------------------------------
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输入 :
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funp, funq, funr 被积分函数系数
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x0 初值,2x2, 按列a, b
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Nn 积分步数
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输出 :
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sol 解矩阵, 2x(Nn+1)
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err 解出标志：false-未解出或达到步数上限；
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true-全部解出
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------------------------------------------------------
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*/
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package goNum
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// ODEDiff 差分方法求解常微分方程
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func ODEDiff(funp, funq, funr func(float64) float64, x0 Matrix, Nn int) (Matrix, bool) {
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/*
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差分方法求解常微分方程
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输入 :
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funp, funq, funr 被积分函数系数
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x0 初值,2x2, 按列a, b
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Nn 积分步数
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输出 :
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sol 解矩阵, 2x(Nn+1)
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err 解出标志：false-未解出或达到步数上限；
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true-全部解出
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*/
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//判断x0维数
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if (x0.Rows != 2) || (x0.Columns != 2) {
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panic("Error in goNum.ODEDiff: Initial values error")
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}
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//判断Nn
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if Nn < 1 {
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panic("Error in goNum.ODEDiff: Nn must greater than zero")
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} else if Nn == 1 {
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return x0, true
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}
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sol := ZeroMatrix(2, Nn+1)
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var err bool = false
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Aa := ZeroMatrix(Nn-1, Nn-1)
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Bb := ZeroMatrix(Nn-1, 1)
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h := (x0.GetFromMatrix(0, 1) - x0.GetFromMatrix(0, 0)) / float64(Nn)
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//ti
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for i := 0; i < Nn+1; i++ {
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sol.SetMatrix(0, i, x0.GetFromMatrix(0, 0)+h*float64(i))
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}
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//x0, xN
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sol.SetMatrix(1, 0, x0.GetFromMatrix(1, 0))
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sol.SetMatrix(1, Nn, x0.GetFromMatrix(1, 1))
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//第一行
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Aa.SetMatrix(0, 0, 2.0+h*h*funq(sol.GetFromMatrix(0, 1)))
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Aa.SetMatrix(0, 1, h*funp(sol.GetFromMatrix(0, 1))/2.0-1.0)
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e0 := (h*funp(sol.GetFromMatrix(0, 1))/2.0 + 1.0) * sol.GetFromMatrix(1, 0)
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Bb.SetMatrix(0, 0, -1.0*h*h*funr(sol.GetFromMatrix(0, 1))+e0)
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for i := 1; i < Nn-2; i++ {
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Aa.SetMatrix(i, i-1, -1.0*h*funp(sol.GetFromMatrix(0, i+1))/2.0-1.0)
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Aa.SetMatrix(i, i, 2.0+h*h*funq(sol.GetFromMatrix(0, i+1)))
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Aa.SetMatrix(i, i+1, h*funp(sol.GetFromMatrix(0, i+1))/2.0-1.0)
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Bb.SetMatrix(i, 0, -1.0*h*h*funr(sol.GetFromMatrix(0, i+1)))
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}
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//最后行
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Aa.SetMatrix(Nn-2, Nn-2-1, -1.0*h*funp(sol.GetFromMatrix(0, Nn-1))/2.0-1.0)
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Aa.SetMatrix(Nn-2, Nn-2, 2.0+h*h*funq(sol.GetFromMatrix(0, Nn-1)))
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eN := (-1.0*h*funp(sol.GetFromMatrix(0, Nn-1))/2.0 + 1.0) * sol.GetFromMatrix(1, Nn)
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Bb.SetMatrix(Nn-2, 0, -1.0*h*h*funr(sol.GetFromMatrix(0, Nn-1))+eN)
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//求解线性方程组LEs_Chasing
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xTemp, errTemp := LEs_Chasing(Aa, Bb)
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if errTemp != true {
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panic("Error in goNum.ODEDiff: Solve error")
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}
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//xTemp赋予sol
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for i := 1; i < Nn; i++ {
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sol.SetMatrix(1, i, xTemp.GetFromMatrix(i-1, 0))
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}
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err = true
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return sol, err
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}
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