引言

随机函数方程在物理、金融、生物等领域的应用逐渐增多，因此对于随机方程的研究非常有必要。由于这类方程的精确解在许多情况下是无法得到的，因此通过一些数值方法来求取近似解是常用且有效的方法。许多学者求解这类方程用不同的函数或多项式，如block-pulse函数^[1]、泰勒级数^[2]、欧拉多项式^[3]、伯努利多项式^[4]和切比雪夫小波^[5-6]等。Fakhrodin^[7]研究了利用Haar小波求解线性随机Ito-Volterra积分方程，但并未给出非线性情况下的数值解。本文在文献[7]基础上，考虑用Haar小波来解非线性随机Ito-Volterra积分方程, 并分析了基于Haar小波的数值解和精确解的误差。

1 Haar小波 1.1 Haar小波结构

定义1^[7-8]  在支撑域为t∈[0, 1)的范围内，Haar函数的定义为

$ \psi (t) = \left\{ {\begin{array}{*{20}{l}} {1, }&{0 \le t < \frac{1}{2}}\\ { - 1, }&{\frac{1}{2} \le t < 1} \end{array}} \right. $

在多分辨率为2^j的系统中，Haar小波构成一组支撑为$\left[{\frac{k}{{{2^j}}}, \frac{{k + 1}}{{{2^j}}}} \right)$的正交小波族

$ {h_n}(t) = {2^{\frac{1}{2}}}\psi \left( {{2^j}t - k} \right), {h_0}(t) = 1 $

式中，$j \ge 0, 0 \le k < {2^j}, n = {2^j} + k, n, j, k \in \mathit{\boldsymbol{Z}}$，且有$\int_0^1 {{h_i}} (t){h_j}(t){\rm{d}}t = {{\rm{ \mathit{ δ} }}_{ij}}$，其中${{\rm{ \mathit{ δ} }}_{ij}} = \left\{ {\begin{array}{*{20}{l}} {1, }&{i = j}\\ {0, }&{i \ne j} \end{array}} \right.$为克罗内克函数。

在区间[0, 1)上定义的平方可积函数f(t)可被Haar小波展开为

$ f(t) = \sum\limits_{i = 0}^\infty {{f_i}} {h_i}(t) $

(1)

式中，$i = {2^j} + k, j \ge 0, 0 \le k < {2^j}, j, k \in {\bf{N}}$，且有$\int_0^1 f (t){h_i}(t){\rm{d}}t$，其中i=0, 2^j+k，j≥0，0≤k < 2^j，j, k∈Ν。

式(1)中的无限序列可被截断为m=2^J(J为给定的小波分辨率水平)，即

$ f(t) \simeq \sum\limits_{i = 0}^{m - 1} {{f_i}} {h_i}(t) $

(2)

式中，i=0, 2^j+k，0≤j≤J－1，0≤k < 2^j。

式(2)用向量形式可表示为

$ f(t) \simeq {\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{H}}(t) = {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{F}} $

式中F和H(t)分别为Haar系数和Haar小波向量，且有

$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{F}} = {{\left[ {{f_0}(t), {f_1}(t), \cdots , {f_{m - 1}}(t)} \right]}^{\rm{T}}}}\\ {\mathit{\boldsymbol{H}}(t) = {{\left[ {{h_0}(t), {h_1}(t), \cdots , {h_{m - 1}}(t)} \right]}^{\rm{T}}}} \end{array}} \right. $

(3)

令k(s, t)∈L²([0, 1)×[0, 1))，类似地k(s, t)可用Haar小波展开为

$ k(s, t) \simeq {\mathit{\boldsymbol{H}}^{\rm{T}}}(s)\mathit{\boldsymbol{KH}}(t) = {\mathit{\boldsymbol{H}}^{\rm{T}}}(t){\mathit{\boldsymbol{K}}^{\rm{T}}}\mathit{\boldsymbol{H}}(s) $

式中K是m×m的Haar小波系数矩阵，其i行l列元素为${k_{il}} = \int_0^1 {\int_0^1 k } (s, t){\mathit{\boldsymbol{H}}_i}(t){\mathit{\boldsymbol{H}}_l}(s){\rm{d}}t{\rm{d}}s, i, l = 1, 2, \cdots, m$。

1.2 block-pulse函数

定义2^{[1, 9]}  定义m个区间的block-pulse函数为

$ \left\{ \begin{array}{l} {\phi _i}(t){\rm{ = }}\left\{ {\begin{array}{*{20}{l}} {1, }&{(i - 1)h \le t < ih}\\ {0, }&{其他{\rm{ }}} \end{array}} \right.\\ {\phi _j}(t) = \left\{ {\begin{array}{*{20}{l}} {1, }&{(j - 1)h \le t < jh}\\ {0, }&{{\rm{ 其他 }}} \end{array}} \right. \end{array} \right. $

(4)

式中，$t \in [0, T), i, j = 1, 2, \cdots, m, h = \frac{T}{m}$，并且${\phi _i}(t)$和${\phi _j}(t)$是不相交的，即

$ {\phi _i}(t){\phi _j}(t) = {\delta _{ij}}{\phi _i}(t), i, j = 1, 2, \cdots , m $

(5)

式(5)中，${\phi _i}(t)$和${\phi _j}(t)$是正交的，即

$ \int_0^T {{\phi _i}} (t){\phi _j}(t){\rm{d}}t = h{\delta _{ij}}, i, j = 1, 2, \cdots , m $

定义2中，若m→∞，则block-pulse函数集$\left\{ {{\phi _i}(t)} \right\}_{i = 1}^\infty $是完备的，所以任意的f∈L²([0, T))可被block-pulse函数序列展开为

$ f(t) \simeq \sum\limits_{i = 1}^m {{f_i}} {\phi _i}(t) = {\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}(t) = {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}(t)\mathit{\boldsymbol{F}} $

式中

$ \left\{ {\begin{array}{*{20}{l}} {{f_i} = \frac{1}{h}\int_0^T {{\phi _i}} (t)f(t){\rm{d}}t, i = 1, 2, \cdots , m}\\ {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}(t) = {{\left[ {{\phi _1}(t), {\phi _2}(t), \cdots , {\phi _m}(t)} \right]}^{\rm{T}}}}\\ {\mathit{\boldsymbol{F}} = {{\left[ {{f_1}, {f_2}, \cdots , {f_m}} \right]}^{\rm{T}}}} \end{array}} \right. $

(6)

对任意的m维列向量F，很容易得到

$ \mathit{\boldsymbol{ \boldsymbol{\varPhi} }}(t){\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}(t)\mathit{\boldsymbol{F}} = \mathit{\boldsymbol{\tilde F \boldsymbol{\varPhi} }}(t) $

(7)

式中，$\mathit{\boldsymbol{\tilde F}} = {\mathop{\rm diag}\nolimits} (\mathit{\boldsymbol{F}})$是m×m矩阵。对任意的m×m矩阵A有

$ {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}(t)\mathit{\boldsymbol{A \boldsymbol{\varPhi} }}(t) = {{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}(t) $

(8)

式中$\mathit{\boldsymbol{\tilde A}} = {\mathop{\rm diag}\nolimits} (\mathit{\boldsymbol{A}})$为m维列向量。

1.3 Haar小波和block-pulse函数关系

令block-pulse函数中的T=1，则会得到Haar小波和block-pulse函数关系。

定义3^[7]  因H(t)和Φ(t)分别为m维Haar小波向量和block-pulse函数向量，相应地，向量H(t)可用向量Φ(t)表示为

$ \mathit{\boldsymbol{H}}(t) = \mathit{\boldsymbol{Q \boldsymbol{\varPhi} }}(t), m = {2^j} $

式中，Q为m×m矩阵，其i行l列元素为

$ {Q_{it}} = {2^{\frac{1}{2}}}{h_{i - 1}}\left( {\frac{{2l - 1}}{{2m}}} \right) $

式中i, l=1, 2，…, m，i－1=2^j+k，0≤k < 2^j。

由式(7)、(8)和定义3很容易得到引理1。

引理1^[10]  对任意的m维列向量F，有

$ \mathit{\boldsymbol{H}}(t){\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{F}} = \mathit{\boldsymbol{\tilde FH}}(t) $

(9)

式中$\mathit{\boldsymbol{\tilde F}} = \mathit{\boldsymbol{Q}}\mathit{\boldsymbol{\overline F}} {\mathit{\boldsymbol{Q}}^{ - 1}}$是m×m矩阵，$\mathit{\boldsymbol{\overline F}} = {\mathop{\rm diag}\nolimits} \left({{\mathit{\boldsymbol{Q}}^{\rm{T}}}\mathit{\boldsymbol{F}}} \right)$。对任意的m×m矩阵A有

$ {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{AH}}(t) = {{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}}\mathit{\boldsymbol{H}}(t) $

(10)

式中${{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}} = \mathit{\boldsymbol{D}}{\mathit{\boldsymbol{Q}}^{ - 1}}, \mathit{\boldsymbol{D}} = {\mathop{\rm diag}\nolimits} \left({{\mathit{\boldsymbol{Q}}^{\rm{T}}}\mathit{\boldsymbol{AQ}}} \right)$。

2 随机积分算子矩阵

与式(6)中Φ(t)有关的积分可表示为^[1]

$ \int_0^t \mathit{\boldsymbol{ \boldsymbol{\varPhi} }} (s){\rm{d}}s \simeq \mathit{\boldsymbol{P \boldsymbol{\varPhi} }}(t) $

(11)

式中P是m×m维block-pulse函数的积分算子矩阵，并且有

$ \mathit{\boldsymbol{P}} = \frac{h}{2}{\left[ {\begin{array}{*{20}{c}} 1&2&2& \cdots &2\\ 0&1&2& \cdots &2\\ 0&0&1& \vdots & \vdots \\ \vdots & \vdots & \vdots & \ddots &2\\ 0&0&0& \cdots &1 \end{array}} \right]_{m \times m}} $

与Φ(t)有关的Ito积分可表示为^[1]率

$ \int_0^t \mathit{\boldsymbol{ \boldsymbol{\varPhi} }} (s){\rm{d}}B(s) \simeq {\mathit{\boldsymbol{P}}_s}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}(t) $

(12)

式中B(s)是布朗运动，P_s是block-pulse函数的随机积分算子矩阵，且有

$ {\mathit{\boldsymbol{P}}_s} = {\left[ {\begin{array}{*{20}{c}} {B\left( {\frac{h}{2}} \right)}&{B(h)}&{B(h)}& \cdots &{B(h)}\\ 0&{B\left( {\frac{{3h}}{2}} \right) - B(h)}&{B(2h) - B(h)}& \cdots &{B(2h) - B(h)}\\ 0&0&{B\left( {\frac{{5h}}{2}} \right) - B(2h)}& \cdots &{B(3h) - B(2h)}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0&0&0& \cdots &{B\left( {\frac{{(2m - 1)h}}{2}} \right) - B((m - 1)h)} \end{array}} \right]_{m \times m}} $

在block-pulse函数的积分算子矩阵基础上，利用定义3可求得Haar小波的算子矩阵。

式(3)中Haar小波向量H(t)的积分可表示为

$ \begin{array}{l} \int_0^t \mathit{\boldsymbol{H}} (s){\rm{d}}s \simeq \int_0^t \mathit{\boldsymbol{Q}} \mathit{\boldsymbol{ \boldsymbol{\varPhi} }}(s){\rm{d}}s = \mathit{\boldsymbol{Q}}\int_0^t \mathit{\boldsymbol{ \boldsymbol{\varPhi} }} (s){\rm{d}}s \simeq \\ \mathit{\boldsymbol{QP \boldsymbol{\varPhi} }}(t) = \mathit{\boldsymbol{QP}}{\mathit{\boldsymbol{Q}}^{ - 1}}\mathit{\boldsymbol{H}}(t) = \mathit{\boldsymbol{AH}}(t) \end{array} $

(13)

式中Λ=QPQ^－1为Haar小波的积分算子矩阵。

同理，H(t)的Ito积分可以表示为

$ \int_0^t \mathit{\boldsymbol{H}} (s){\rm{d}}B(s) \simeq \mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{P}}_\mathit{s}}{\mathit{\boldsymbol{Q}}^{ - 1}}\mathit{\boldsymbol{H}}(t) = {\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_\mathit{s}}\mathit{\boldsymbol{H}}(t) $

(14)

式中${\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_s} = \mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{P}}_s}{\mathit{\boldsymbol{Q}}^{ - 1}}$为Haar小波的随机积分算子矩阵。

3 方程的转化和求解

$ \begin{array}{l} X(t) = f(t) + \int_0^t {{k_1}} (s, t){N_1}(s, X(s)){\rm{d}}s + \\ \int_0^t {{k_2}} (s, t){N_2}(s, X(s)){\rm{d}}B(s), t \in [0, 1] \end{array} $

(15)

式(15)为非线性随机Ito-Volterra积分方程。式中，X(t)、f(t)、k₁(s, t)、k₂(s, t)、N₁(s, X(s))和N₂(s, X(s))是定义在概率空间(Ω, F, P)上的随机过程，且X(t)未知。

利用Haar小波随机积分算子矩阵求解式(15)。首先令

$ \left\{ {\begin{array}{*{20}{l}} {{u_1}(t) = {N_1}(t, X(t))}\\ {{u_2}(t) = {N_2}(t, X(t))} \end{array}} \right. $

(16)

由式(15)和式(16)，可得

$ \left\{ {\begin{array}{*{20}{c}} {{u_1}(t) = {N_1}\left( {t, f(t) + \int_0^t {{k_1}} (s, t){u_1}(s){\rm{d}}s + } \right.}\\ {\left. {\int_0^t {{k_2}} (s, t){u_2}(s){\rm{d}}B(s)} \right)}\\ {{u_2}(t) = {N_2}\left( {t, f(t) + \int_0^t {{k_1}} (s, t){u_1}(s){\rm{d}}s + } \right.}\\ {\left. {\int_0^t {{k_2}} (s, t){u_2}(s){\rm{d}}B(s)} \right)} \end{array}} \right. $

(17)

然后将X(t)、f(t)、k₁(s, t)、k₂(s, t)、u₁(t)和u₂(t)近似用Haar小波向量表示为

$ \left\{ {\begin{array}{*{20}{l}} {X(t) \simeq {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{H}}(t) = {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{X}}}\\ {f(t) \simeq {\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{H}}(t) = {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{F}}}\\ {{k_1}(s, t) \simeq {\mathit{\boldsymbol{H}}^{\rm{T}}}(s){\mathit{\boldsymbol{K}}_1}\mathit{\boldsymbol{H}}(t) = {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{K}}_1^{\rm{T}}\mathit{\boldsymbol{H}}(s)}\\ {{k_2}(s, t) \simeq {\mathit{\boldsymbol{H}}^{\rm{T}}}(s){\mathit{\boldsymbol{K}}_2}\mathit{\boldsymbol{H}}(t) = {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{K}}_2^{\rm{T}}\mathit{\boldsymbol{H}}(s)}\\ {{u_1}(t) \simeq \mathit{\boldsymbol{U}}_1^{\rm{T}}\mathit{\boldsymbol{H}}(t) = {\mathit{\boldsymbol{H}}^{\rm{T}}}(t){\mathit{\boldsymbol{U}}_1}}\\ {{u_2}(t) \simeq \mathit{\boldsymbol{U}}_2^{\rm{T}}\mathit{\boldsymbol{H}}(t) = {\mathit{\boldsymbol{H}}^{\rm{T}}}(t){\mathit{\boldsymbol{U}}_2}} \end{array}} \right. $

(18)

将式(18)的近似趋近代入式(17)可得

$ \left\{ \begin{array}{l} {\mathit{\boldsymbol{H}}^{\rm{T}}}(t){\mathit{\boldsymbol{U}}_1} = {N_1}\left( {t, {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{F}} + } \right.\\ \;\;\;\;\int_0^t {{\mathit{\boldsymbol{H}}^{\rm{T}}}} (t)\mathit{\boldsymbol{K}}_1^{\rm{T}}\mathit{\boldsymbol{H}}(s){\mathit{\boldsymbol{H}}^{\rm{T}}}(s){\mathit{\boldsymbol{U}}_1}{\rm{d}}s + \\ \;\;\;\;\left. {\int_0^t {{\mathit{\boldsymbol{H}}^{\rm{T}}}} (t)\mathit{\boldsymbol{K}}_2^{\rm{T}}\mathit{\boldsymbol{H}}(s){\mathit{\boldsymbol{H}}^{\rm{T}}}(s){\mathit{\boldsymbol{U}}_2}{\rm{d}}B(\mathit{s})} \right)\\ {\mathit{\boldsymbol{H}}^{\rm{T}}}(t){\mathit{\boldsymbol{U}}_2} = {N_2}\left( {t, {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{F}} + } \right.\\ \;\;\;\;\int_0^t {{\mathit{\boldsymbol{H}}^{\rm{T}}}} (t)\mathit{\boldsymbol{K}}_1^{\rm{T}}\mathit{\boldsymbol{H}}(s){\mathit{\boldsymbol{H}}^{\rm{T}}}(s){\mathit{\boldsymbol{U}}_1}{\rm{d}}s + \\ \left. {\;\;\;\;\int_0^t {{\mathit{\boldsymbol{H}}^{\rm{T}}}} (t)\mathit{\boldsymbol{K}}_2^{\rm{T}}\mathit{\boldsymbol{H}}(s){\mathit{\boldsymbol{H}}^{\rm{T}}}(s){\mathit{\boldsymbol{U}}_2}{\rm{d}}\mathit{B}(s)} \right) \end{array} \right. $

(19)

通过式(8)、(9)、(13)和(14)，式(19)可化为

$ \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{H}}^{\rm{T}}}(t){\mathit{\boldsymbol{U}}_1} = {N_1}\left( {t, {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{F}} + \mathit{\boldsymbol{\tilde V}}_1^{\rm{T}}\mathit{\boldsymbol{H}}(t) + \mathit{\boldsymbol{\tilde V}}_2^{\rm{T}}\mathit{\boldsymbol{H}}(t)} \right)}\\ {{\mathit{\boldsymbol{H}}^{\rm{T}}}(t){\mathit{\boldsymbol{U}}_2} = {N_2}\left( {t, {\mathit{\boldsymbol{H}}^{\rm{T}}}(t)\mathit{\boldsymbol{F}} + \mathit{\boldsymbol{\tilde V}}_1^{\rm{T}}\mathit{\boldsymbol{H}}(t) + \mathit{\boldsymbol{\tilde V}}_2^{\rm{T}}\mathit{\boldsymbol{H}}(t)} \right)} \end{array}} \right. $

式中

$ \begin{array}{l} \mathit{\boldsymbol{\tilde V}}_1^{\rm{T}} = {\mathit{\boldsymbol{D}}_1}{\mathit{\boldsymbol{Q}}^{ - 1}},{\mathit{\boldsymbol{D}}_1} = {\mathop{\rm diag}\nolimits} \left( {{\mathit{\boldsymbol{Q}}^{\rm{T}}}\mathit{\boldsymbol{K}}_1^{\rm{T}}{{\mathit{\boldsymbol{\tilde U}}}_1}\mathit{\boldsymbol{ \boldsymbol{\varLambda} Q}}} \right)\\ \mathit{\boldsymbol{\tilde V}}_2^{\rm{T}} = {\mathit{\boldsymbol{D}}_2}{\mathit{\boldsymbol{Q}}^{ - 1}},{\mathit{\boldsymbol{D}}_2} = {\mathop{\rm diag}\nolimits} \left( {{\mathit{\boldsymbol{Q}}^{\rm{T}}}\mathit{\boldsymbol{K}}_2^{\rm{T}}{{\mathit{\boldsymbol{\tilde U}}}_2}\mathit{\boldsymbol{A}},\mathit{\boldsymbol{Q}}} \right)\\ {{\mathit{\boldsymbol{\tilde U}}}_i} = \mathit{\boldsymbol{Q}}{\mathop{\rm diag}\nolimits} \left( {{\mathit{\boldsymbol{Q}}^{ - 1}}{\mathit{\boldsymbol{U}}_i}} \right){\mathit{\boldsymbol{Q}}^{ - 1}},i = 1,2 \end{array} $

给出m个牛顿cotes节点${t_i} = \frac{{2i + 1}}{{2m}}$，i=0, 1, …, m-1，则有

$ \left\{ \begin{array}{l} {\mathit{\boldsymbol{H}}^{\rm{T}}}\left( {{t_i}} \right){\mathit{\boldsymbol{U}}_1} = {N_1}\left( {{t_i}, {\mathit{\boldsymbol{H}}^{\rm{T}}}\left( {{t_i}} \right)\mathit{\boldsymbol{F}} + \mathit{\boldsymbol{\widetilde V}}_1^{\rm{T}}\mathit{\boldsymbol{H}}\left( {{t_i}} \right) + \mathit{\boldsymbol{\tilde V}}_2^{\rm{T}}\mathit{\boldsymbol{H}}\left( {{t_i}} \right)} \right)\\ {\mathit{\boldsymbol{H}}^{\rm{T}}}\left( {{t_i}} \right){\mathit{\boldsymbol{U}}_2} = {N_2}\left( {{t_i}, {\mathit{\boldsymbol{H}}^{\rm{T}}}\left( {{t_i}} \right)\mathit{\boldsymbol{F}} + \mathit{\boldsymbol{\widetilde V}}_1^{\rm{T}}\mathit{\boldsymbol{H}}\left( {{t_i}} \right) + } \right.\left. {\mathit{\boldsymbol{\widetilde V}}_2^{\rm{T}}\mathit{\boldsymbol{H}}\left( {{t_i}} \right)} \right) \end{array} \right. $

(20)

式(20)给出了一个有2m个代数方程的非线性系统，每个代数方程都有同样数量的未知系数，这样就可以通过牛顿迭代方法得到U₁和U₂的分量，进而可得到式(15)的近似解为

$ {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{H}}(t) \simeq {\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{H}}(t) + \mathit{\boldsymbol{\widetilde V}}_1^{\rm{T}}\mathit{\boldsymbol{H}}(t) + \mathit{\boldsymbol{\widetilde V}}_2^{\rm{T}}\mathit{\boldsymbol{H}}(t) $

即

$ {\mathit{\boldsymbol{X}}^{\rm{T}}} \simeq {\mathit{\boldsymbol{F}}^{\rm{T}}} + \mathit{\boldsymbol{\widetilde V}}_1^{\rm{T}} + \mathit{\boldsymbol{\widetilde V}}_2^{\rm{T}} $

(21)

4 误差分析

对第3节得到的目标方程的求解方法进行误差分析。将X_m(t)记为由式(21)得到的X(t)的近似解，由于函数的精确解未知，所以利用误差估计e_m(t)=X(t)－X_m(t)来衡量近似解的优劣，并定义||X(t)||=sup|X(t)|。

定理1^[7]  假设f(t)和k(s, t)是足够光滑的函数，$\hat f(t)$和$\hat k(\mathit{s, }t)$分别是f(t)和k(s, t)由Haar小波函数得到的近似估计，则f(t)与$\hat f(t)$、k(s, t)与$\hat k(\mathit{s, }t)$的误差边界为

$ \left\{ {\begin{array}{*{20}{l}} {f(t) - \hat f(t) = O\left( {{m^{ - 1}}} \right), }&{t \in [0, 1]}\\ {k(s, t) - \hat k(s, t) = 0\left( {{m^{ - 1}}} \right), }&{(s, t) \in [0, 1] \times [0, 1]} \end{array}} \right. $

(22)

定理2  假设X(t)和X_m(t)分别为式(15)的精确解和由Haar小波得出的近似解，给出以下条件：

①‖X(t)‖≤r，t∈[0, 1]；

②‖k_i(s, t)‖≤M_i，(s, t)∈[0, 1]×[0, 1]，i=1, 2；

③ Lipschitz条件

‖N_i(t, X(t))－N_i(t, X_m(t))‖≤L_i‖X(t)－X_m(t)‖，i=1, 2；

④ 线性增长条件

‖N_i(t, X(t))‖≤L_i(1+‖X(t)‖)，i=1, 2；

⑤ L₁(M₁+λ₁(m))+‖B(t)‖L₂(M₂+λ₂(m)) < 1。

从而可以得到

$ \left\|e_{m}(t)\right\|=\left\|X(t)-X_{m}(t)\right\|=O\left(m^{-1}\right) $

(23)

式中${\lambda _i}(m) = {{\tilde C}_i}\frac{1}{m}, i = 1, 2, {\rm{ }}{{\tilde C}_i}$为常数。

证明：假设u_i(s)和$\hat{u}_{i}(s)$分别是方程(17)的精确值和近似值，则有

$ \left\{ {\begin{array}{*{20}{l}} {{{\hat u}_i}(s) = {{\hat N}_i}\left( {s, {X_m}(s)} \right), }&{i = 1, 2}\\ {u_i^m(s) = {N_i}\left( {s, {X_m}(s)} \right), }&{i = 1, 2} \end{array}} \right. $

(24)

由条件③和定理1可得

$ \begin{array}{l} \left\| {{u_i}(s) - {{\hat u}_i}(s)} \right\| \le \left\| {{u_i}(s) - u_i^m(s)} \right\| + \\ \left\| {u_i^m(s) - {{\hat u}_i}(s)} \right\| \le {L_i}\left\| {{e_m}(s)} \right\| + {\beta _i}(m) \end{array} $

式中，${\beta _i}(m) = {C_i}\frac{1}{m}, i = 1, 2, {C_i}$为常数。进而有

$ \left\{ {\begin{array}{*{20}{c}} {X(t) = f(t) + \int_0^t {{k_1}} (s, t){u_1}(s){\rm{d}}s + }\\ {\int_0^t {{k_2}} (s, t){u_2}(s){\rm{d}}B(s)}\\ {{X_m}(t) = \hat f(t) + \int_0^t {{{\hat k}_1}} (s, t){{\hat u}_1}(s){\rm{d}}s + }\\ {\int_0^t {{{\hat k}_2}} (s, t){{\hat u}_2}(s){\rm{d}}B(s)} \end{array}} \right. $

(25)

由式(25)可得

$ \begin{array}{l} \left\| {X(t) - {X_m}(t)} \right\| \le f(t) - \hat f(t) + {k_1}(s, \\ t){u_1}(s) - {{\hat k}_1}(s, t){{\mathit{\hat u}}_1}(s) + B(t){k_2}(s, t)\\ {u_2}(s) - {{\hat k}_2}(s, t){{\hat u}_2}(s) \end{array} $

(26)

进一步由定理1和条件①、②、④可得

$ \begin{array}{l} \left\| {{k_i}(s, t){u_i}(s) - {{\hat k}_i}(s, t){{\hat u}_i}(s)} \right\| \le \left\| {{k_i}(s, t)} \right\|\\ \left\| {{u_i}(s) - {{\hat u}_i}(s)} \right\| + \left\| {{k_i}(s, t) - {{\hat k}_i}(s, t)} \right\|\left( {{u_i}(s) - } \right.\\ \left. {{{\hat u}_i}(s) + {u_i}(s)} \right) \le \left( {{M_i} + {\lambda _i}(m)} \right){L_i}\left\| {{e_m}(s)} \right\| + \\ \left( {{M_i} + {\lambda _i}(m)} \right){\beta _i}(m)| + {\lambda _i}(m){{\tilde L}_i}(1 + r) \end{array} $

(27)

由式(26)、(27)和条件⑤可得

$ \left\|e_{m}(t)\right\|=\left\|X(t)-X_{m}(t)\right\|=O\left(m^{-1}\right) $

综上，定理2得证。

5 结束语

本文利用Haar小波的算子矩阵和随机算子矩阵求解非线性随机Ito-Volterra方程，得到了数值解方程，然后通过对目标方法的收敛分析和误差分析得出，基于Haar小波的非线性随机Ito-Volterra积分方程的数值解是非常方便和有效的。