#### 期刊菜单

Similar Construction Method of Boundary Value Problem of Three-Interval Composite Riccati-Bessel Equation

Abstract: By analysing and simplifying the solution expressions of the three-interval composite Riccati-Bessel equation, it is found that the solutions of this type of boundary value problems are in the form of continuous fractions and similarity, which are combined by the induced solution function, similar kernel function, inner and outer boundaries, and articulation condition coefficients. Therefore, a new method to solve this type of three-interval composite Riccati-Bessel equation boundary value problems—“similarity construction method” is obtained. This method greatly simplifies the solution process, and the obtained solution expressions are simple and beautiful.

1. 引言

Riccati-Bessel方程是特殊的Riccati方程，在球体的电磁散射 [11] 研究中Riccati-Bessel方程的解Riccati-Bessel函数有着重要的意义和作用。那么对于Riccati-Bessel方程的求解研究重要性不言而喻。近年来，李顺初等人对Riccati-Bessel方程 [12] 、复合Riccati-Bessel方程 [13] 的求解进行了研究，并得到了解的相似结构，但是尚未对于三区复合型及其以上的Riccati-Bessel方程边值问题解的研究。基于以上基础，本文研究如下的三区复合型Riccati-Bessel方程边值问题

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2. 预备知识

${y}_{i}=\sqrt{\frac{\pi x}{2}}\left[{M}_{i}{Y}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)+{N}_{i}{J}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)\right],$

$\begin{array}{l}{\phi }_{0,0}^{i}\left(x,\xi \right)=\sqrt{\frac{\pi x}{2}}{Y}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)\sqrt{\frac{\pi \xi }{2}}{J}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}\xi \right)-\sqrt{\frac{\pi x}{2}}{J}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)\sqrt{\frac{\pi \xi }{2}}{Y}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}\xi \right)\\ \text{}=\frac{\pi }{2}\sqrt{x\xi }{\psi }_{{l}_{i}+\frac{1}{2},{l}_{i}+\frac{1}{2}}\left(x,\xi ,\frac{{\mu }_{ni}}{{a}_{i}}\right),\end{array}$ (2)

$\begin{array}{l}{\phi }_{0,1}^{i}\left(x,\xi \right)=\frac{\partial }{\partial \xi }{\phi }_{0,0}^{i}\left(x,\xi \right)\\ \text{}=\frac{\pi }{2}\sqrt{\frac{x}{\xi }}\left[\left({l}_{i}+1\right){\psi }_{{l}_{i}+\frac{1}{2},{l}_{i}+\frac{1}{2}}\left(x,\xi ,\frac{{\mu }_{ni}}{{a}_{i}}\right)-\frac{{\mu }_{ni}}{{a}_{i}}\xi {\psi }_{{l}_{i}+\frac{1}{2},{l}_{i}+\frac{3}{2}}\left(x,\xi ,\frac{{\mu }_{ni}}{{a}_{i}}\right)\right],\end{array}$ (3)

$\begin{array}{l}{\phi }_{1,0}^{i}\left(x,\xi \right)=\frac{\partial }{\partial x}{\phi }_{0,0}^{i}\left(x,\xi \right)\\ \text{}=\frac{\pi }{2}\sqrt{\frac{\xi }{x}}\left[\left({l}_{i}+1\right){\psi }_{{l}_{i}+\frac{1}{2},{l}_{i}+\frac{1}{2}}\left(x,\xi ,\frac{{\mu }_{ni}}{{a}_{i}}\right)-\frac{{\mu }_{ni}}{{a}_{i}}x{\psi }_{{l}_{i}+\frac{3}{2},{l}_{i}+\frac{1}{2}}\left(x,\xi ,\frac{{\mu }_{ni}}{{a}_{i}}\right)\right],\end{array}$ (4)

$\begin{array}{l}{\phi }_{1,1}^{i}\left(x,\xi \right)=\frac{{\partial }^{2}}{\partial x\partial \xi }{\phi }_{0,0}^{i}\left(x,\xi \right)\\ \text{}=\frac{\pi }{2}\sqrt{\frac{1}{x\xi }}\left[{\left({l}_{i}+1\right)}^{2}{\psi }_{{l}_{i}+\frac{1}{2},{l}_{i}+\frac{1}{2}}\left(x,\xi ,\frac{{\mu }_{ni}}{{a}_{i}}\right)-\left({l}_{i}+1\right)\frac{{\mu }_{ni}}{{a}_{i}}\xi {\psi }_{{l}_{i}+\frac{1}{2},{l}_{i}+\frac{3}{2}}\left(x,\xi ,\frac{{\mu }_{ni}}{{a}_{i}}\right)\\ \text{}-\left({l}_{i}+1\right)\frac{{\mu }_{ni}}{{a}_{i}}x{\psi }_{{l}_{i}+\frac{3}{2},{l}_{i}+\frac{1}{2}}\left(x,\xi ,\frac{{\mu }_{ni}}{{a}_{i}}\right)+\frac{{\mu }_{ni}^{2}}{{a}_{i}^{2}}x\xi {\psi }_{{l}_{i}+\frac{3}{2},{l}_{i}+\frac{3}{2}}\left(x,\xi ,\frac{{\mu }_{ni}}{{a}_{i}}\right)\right].\end{array}$ (5)

${\psi }_{m,n}\left(x,y,t\right)={Y}_{m}\left(xt\right){J}_{n}\left(yt\right)-{J}_{m}\left(xt\right){Y}_{n}\left(yt\right).$

3. 主要定理及证明

${y}_{1}=D\cdot \frac{1}{E+\frac{1}{F+{\Phi }_{1}\left(a\right)}}\cdot \frac{1}{F+{\Phi }_{1}\left(a\right)}\cdot {\Phi }_{1}\left(x\right),$ (6)

${y}_{2}=D\cdot \frac{1}{E+\frac{1}{F+{\Phi }_{1}\left(a\right)}}\cdot \frac{1}{F+{\Phi }_{1}\left(a\right)}\cdot \frac{{\phi }_{0,1}^{1}\left(b,b\right)}{{\gamma }_{1}{\Phi }_{2}\left(b\right){\phi }_{1,1}^{1}\left(a,b\right)-{\gamma }_{2}{\phi }_{1,0}^{1}\left(a,b\right)}\cdot {\Phi }_{2}\left(x\right),$ (7)

$\begin{array}{l}{y}_{3}=D\cdot \frac{1}{E+\frac{1}{F+{\Phi }_{1}\left(a\right)}}\cdot \frac{1}{F+{\Phi }_{1}\left(a\right)}\cdot \frac{{\phi }_{0,1}^{1}\left(b,b\right)}{{\gamma }_{1}{\Phi }_{2}\left(b\right){\phi }_{1,1}^{1}\left(a,b\right)-{\gamma }_{2}{\phi }_{1,0}^{1}\left(a,b\right)}\\ \text{}\cdot \frac{{\phi }_{0,1}^{2}\left(c,c\right)}{{\eta }_{1}{\Phi }_{3}\left(c\right){\phi }_{1,1}^{2}\left(b,c\right)-{\eta }_{2}{\phi }_{1,0}^{2}\left(b,c\right)}{\Phi }_{3}\left(x\right),\end{array}$ (8)

${\Phi }_{3}\left(x\right)=\frac{G{\phi }_{0,0}^{3}\left(x,d\right)+H{\phi }_{0,1}^{3}\left(x,d\right)}{G{\phi }_{1,0}^{3}\left(c,d\right)+H{\phi }_{1,1}^{3}\left(c,d\right)},\left(c\le x\le d\right)$ (9)

${\Phi }_{2}\left(x\right)$ 称为中区相似核函数

${\Phi }_{2}\left(x\right)=\frac{{\eta }_{2}{\phi }_{0,0}^{2}\left(x,c\right)-{\eta }_{1}{\Phi }_{3}\left(c\right){\phi }_{0,1}^{2}\left(x,c\right)}{{\eta }_{2}{\phi }_{1,0}^{2}\left(b,c\right)-{\eta }_{1}{\Phi }_{3}\left(c\right){\phi }_{1,1}^{2}\left(b,c\right)},\left(b\le x\le c\right)$ (10)

${\Phi }_{1}\left(x\right)$ 称为内区相似核函数

${\Phi }_{1}\left(x\right)=\frac{{\gamma }_{2}{\phi }_{0,0}^{1}\left(x,b\right)-{\gamma }_{1}{\Phi }_{2}\left(b\right){\phi }_{0,1}^{1}\left(x,b\right)}{{\gamma }_{2}{\phi }_{1,0}^{1}\left(a,b\right)-{\gamma }_{1}{\Phi }_{2}\left(b\right){\phi }_{1,1}^{1}\left(a,b\right)},\left(a\le x\le b\right)$ (11)

${y}_{i}=\sqrt{\frac{\pi x}{2}}\left[{M}_{i}{Y}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)+{N}_{i}{J}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)\right],\left(i=1,2,3\right),$ (12)

$\begin{array}{l}{y}_{i}{}^{\prime }={M}_{i}\sqrt{\frac{\pi }{2}}\left[\left({l}_{i}+1\right){x}^{-\frac{1}{2}}{Y}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)-{x}^{\frac{1}{2}}\frac{{\mu }_{ni}}{{a}_{i}}{Y}_{{l}_{i}+\frac{3}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)\right]\\ \text{}+{N}_{i}\sqrt{\frac{\pi }{2}}\left[\left({l}_{i}+1\right){x}^{-\frac{1}{2}}{J}_{{l}_{i}+\frac{1}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)-{x}^{\frac{1}{2}}\frac{{\mu }_{ni}}{{a}_{i}}{J}_{{l}_{i}+\frac{3}{2}}\left(\frac{{\mu }_{ni}}{{a}_{i}}x\right)\right].\end{array}$ (13)

$\begin{array}{l}{M}_{1}\sqrt{\frac{\pi }{2}}\left\{\left[E{a}^{\frac{1}{2}}+\left(1+EF\right)\left({l}_{1}+1\right){a}^{-\frac{1}{2}}\right]{Y}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}a\right)-\left(1+EF\right){a}^{\frac{1}{2}}\frac{{\mu }_{n1}}{{a}_{1}}{Y}_{{l}_{1}+\frac{3}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}a\right)\right\}\\ +\text{\hspace{0.17em}}{N}_{1}\sqrt{\frac{\pi }{2}}\left\{\left[E{a}^{\frac{1}{2}}+\left(1+EF\right)\left({l}_{1}+1\right){a}^{-\frac{1}{2}}\right]{J}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}a\right)-\left(1+EF\right){a}^{\frac{1}{2}}\frac{{\mu }_{n1}}{{a}_{1}}{J}_{{l}_{1}+\frac{3}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}a\right)\right\}=D.\end{array}$ (14)

${M}_{1}{Y}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)+{N}_{2}{J}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)-{M}_{2}{\gamma }_{1}{Y}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}b\right)-{N}_{2}{\gamma }_{1}{J}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}b\right)=0,$ (15)

$\begin{array}{l}{M}_{1}\left[\left({l}_{1}+1\right){b}^{-\frac{1}{2}}{Y}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)-{b}^{\frac{1}{2}}\frac{{\mu }_{n1}}{{a}_{1}}{Y}_{{l}_{1}+\frac{3}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)\right]\\ +\text{\hspace{0.17em}}{N}_{1}\left[\left({l}_{1}+1\right){b}^{-\frac{1}{2}}{J}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)-{b}^{\frac{1}{2}}\frac{{\mu }_{n1}}{{a}_{1}}{J}_{{l}_{1}+\frac{3}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)\right]\\ -\text{\hspace{0.17em}}{M}_{2}{\gamma }_{2}\left[\left({l}_{2}+1\right){b}^{-\frac{1}{2}}{Y}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}b\right)-{b}^{\frac{1}{2}}\frac{{\mu }_{n2}}{{a}_{2}}{Y}_{{l}_{2}+\frac{3}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}b\right)\right]\\ -\text{\hspace{0.17em}}{N}_{2}{\gamma }_{2}\left[\left({l}_{2}+1\right){b}^{-\frac{1}{2}}{J}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}b\right)-{b}^{\frac{1}{2}}\frac{{\mu }_{n2}}{{a}_{2}}{J}_{{l}_{2}+\frac{3}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}b\right)\right]=0,\end{array}$ (16)

 (17)

$\begin{array}{l}{M}_{2}\left[\left({l}_{2}+1\right){c}^{-\frac{1}{2}}{Y}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)-{c}^{\frac{1}{2}}\frac{{\mu }_{n2}}{{a}_{2}}{Y}_{{l}_{2}+\frac{3}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)\right]\\ +\text{\hspace{0.17em}}{N}_{2}\left[\left({l}_{2}+1\right){c}^{-\frac{1}{2}}{J}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)-{c}^{\frac{1}{2}}\frac{{\mu }_{n2}}{{a}_{2}}{J}_{{l}_{2}+\frac{3}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)\right]\\ -\text{\hspace{0.17em}}{M}_{3}{\eta }_{2}\left[\left({l}_{3}+1\right){c}^{-\frac{1}{2}}{Y}_{{l}_{3}+\frac{1}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}c\right)-{c}^{\frac{1}{2}}\frac{{\mu }_{n3}}{{a}_{3}}{Y}_{{l}_{3}+\frac{3}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}c\right)\right]\\ -\text{\hspace{0.17em}}{N}_{3}{\eta }_{2}\left[\left({l}_{3}+1\right){c}^{-\frac{1}{2}}{J}_{{l}_{3}+\frac{1}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}c\right)-{c}^{\frac{1}{2}}\frac{{\mu }_{n3}}{{a}_{3}}{J}_{{l}_{3}+\frac{3}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}c\right)\right]=0.\end{array}$ (18)

$\begin{array}{l}{M}_{3}\left\{\left[G{d}^{\frac{1}{2}}+H\left({l}_{3}+1\right){d}^{-\frac{1}{2}}\right]{Y}_{{l}_{3}+\frac{1}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)-H{d}^{\frac{1}{2}}\frac{{\mu }_{n3}}{{a}_{3}}{Y}_{{l}_{3}+\frac{3}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)\right\}\\ +\text{\hspace{0.17em}}{N}_{3}\left\{\left[G{d}^{\frac{1}{2}}+H\left({l}_{3}+1\right){d}^{-\frac{1}{2}}\right]{J}_{{l}_{3}+\frac{1}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)-H{d}^{\frac{1}{2}}\frac{{\mu }_{n3}}{{a}_{3}}{J}_{{l}_{3}+\frac{3}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)\right\}=0.\end{array}$ (19)

$\begin{array}{c}\Delta ={\left(bc\right)}^{-\frac{1}{2}}{\left(\frac{\pi }{2}\right)}^{-\frac{5}{2}}\left\{{\gamma }_{2}{\eta }_{2}{\phi }_{1,0}^{2}\left(b,c\right)\left[E{\phi }_{0,0}^{1}\left(a,b\right)+\left(1+EF\right){\phi }_{1,0}^{1}\left(a,b\right)\right]\left[G{\phi }_{1,0}^{3}\left(c,d\right)+H{\phi }_{1,1}^{3}\left(c,d\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\gamma }_{1}{\eta }_{2}{\phi }_{0,0}^{2}\left(b,c\right)\left[E{\phi }_{0,1}^{1}\left(a,b\right)+\left(1+EF\right){\phi }_{1,1}^{1}\left(a,b\right)\right]\left[G{\phi }_{1,0}^{3}\left(c,d\right)+H{\phi }_{1,1}^{3}\left(c,d\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\gamma }_{2}{\eta }_{1}{\phi }_{1,1}^{2}\left(b,c\right)\left[E{\phi }_{0,0}^{1}\left(a,b\right)+\left(1+EF\right){\phi }_{1,0}^{1}\left(a,b\right)\right]\left[G{\phi }_{0,0}^{3}\left(c,d\right)+H{\phi }_{0,1}^{3}\left(c,d\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\gamma }_{1}{\eta }_{1}{\phi }_{0,1}^{2}\left(b,c\right)\left[E{\phi }_{0,1}^{1}\left(a,b\right)+\left(1+EF\right){\phi }_{1,1}^{1}\left(a,b\right)\right]\left[G{\phi }_{0,0}^{3}\left(c,d\right)+H{\phi }_{0,1}^{3}\left(c,d\right)\right]\right\}.\end{array}$ (20)

$\begin{array}{c}{M}_{1}=\frac{D}{\Delta }{\left(\frac{\pi }{2}\right)}^{-2}{c}^{-\frac{1}{2}}\left\{{\gamma }_{2}{\eta }_{2}{J}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right){\phi }_{1,0}^{2}\left(b,c\right)\left[G{\phi }_{1,0}^{3}\left(c,d\right)+H{\phi }_{1,1}^{3}\left(c,d\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\gamma }_{2}{\eta }_{1}{J}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right){\phi }_{1,1}^{2}\left(b,c\right)\left[G{\phi }_{0,0}^{3}\left(c,d\right)+H{\phi }_{0,1}^{3}\left(c,d\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\gamma }_{1}{\eta }_{2}{b}^{-\frac{1}{2}}{\phi }_{0,0}^{2}\left(b,c\right)\left[\left({l}_{1}+1\right){b}^{-\frac{1}{2}}{J}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)-{b}^{\frac{1}{2}}\frac{{\mu }_{n1}}{{a}_{1}}{J}_{{l}_{1}+\frac{3}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)\right]\left[G{\phi }_{1,0}^{3}\left(c,d\right)+H{\phi }_{1,1}^{3}\left(c,d\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\gamma }_{1}{\eta }_{1}{b}^{-\frac{1}{2}}{\phi }_{0,1}^{2}\left(b,c\right)\left[\left({l}_{1}+1\right){b}^{-\frac{1}{2}}{J}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)-{b}^{\frac{1}{2}}\frac{{\mu }_{n1}}{{a}_{1}}{J}_{{l}_{1}+\frac{3}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)\right]\left[G{\phi }_{0,0}^{3}\left(c,d\right)+H{\phi }_{0,1}^{3}\left(c,d\right)\right]\right\},\end{array}$ (21)

$\begin{array}{c}{N}_{1}=-\frac{D}{\Delta }{\left(\frac{\pi }{2}\right)}^{-2}{c}^{-\frac{1}{2}}\left\{{\gamma }_{2}{\eta }_{2}{Y}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right){\phi }_{1,0}^{2}\left(b,c\right)\left[G{\phi }_{1,0}^{3}\left(c,d\right)+H{\phi }_{1,1}^{3}\left(c,d\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\gamma }_{2}{\eta }_{1}{Y}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right){\phi }_{1,1}^{2}\left(b,c\right)\left[G{\phi }_{0,0}^{3}\left(c,d\right)+H{\phi }_{0,1}^{3}\left(c,d\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\gamma }_{1}{\eta }_{2}{b}^{-\frac{1}{2}}{\phi }_{0,0}^{2}\left(b,c\right)\left[\left({l}_{1}+1\right){b}^{-\frac{1}{2}}{Y}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)-{b}^{\frac{1}{2}}\frac{{\mu }_{n1}}{{a}_{1}}{Y}_{{l}_{1}+\frac{3}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)\right]\left[G{\phi }_{1,0}^{3}\left(c,d\right)+H{\phi }_{1,1}^{3}\left(c,d\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\gamma }_{1}{\eta }_{1}{b}^{-\frac{1}{2}}{\phi }_{0,1}^{2}\left(b,c\right)\left[\left({l}_{1}+1\right){b}^{-\frac{1}{2}}{Y}_{{l}_{1}+\frac{1}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)-{b}^{\frac{1}{2}}\frac{{\mu }_{n1}}{{a}_{1}}{Y}_{{l}_{1}+\frac{3}{2}}\left(\frac{{\mu }_{n1}}{{a}_{1}}b\right)\right]\left[G{\phi }_{0,0}^{3}\left(c,d\right)+H{\phi }_{0,1}^{3}\left(c,d\right)\right]\right\},\end{array}$ (22)

$\begin{array}{l}{M}_{2}=-\frac{D}{\Delta }{\left(\frac{\pi }{2}\right)}^{-2}{b}^{-\frac{1}{2}}{\phi }_{0,1}^{1}\left(b,b\right)\left\{{\eta }_{2}{J}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)\left[G{\phi }_{1,0}^{3}\left(c,d\right)+H{\phi }_{1,1}^{3}\left(c,d\right)\right]\\ \text{}-{\eta }_{1}{c}^{-\frac{1}{2}}\left[\left({l}_{2}+1\right){c}^{-\frac{1}{2}}{J}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)-{c}^{\frac{1}{2}}\frac{{\mu }_{n2}}{{a}_{2}}{J}_{{l}_{2}+\frac{3}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)\right]\left[G{\phi }_{0,0}^{3}\left(c,d\right)+H{\phi }_{0,1}^{3}\left(c,d\right)\right]\right\},\end{array}$ (23)

$\begin{array}{l}{N}_{2}=\frac{D}{\Delta }{\left(\frac{\pi }{2}\right)}^{-2}{b}^{-\frac{1}{2}}{\phi }_{0,1}^{1}\left(b,b\right)\left\{{\eta }_{2}{Y}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)\left[G{\phi }_{1,0}^{3}\left(c,d\right)+H{\phi }_{1,1}^{3}\left(c,d\right)\right]\\ \text{}-{\eta }_{1}{c}^{-\frac{1}{2}}\left[\left({l}_{2}+1\right){c}^{-\frac{1}{2}}{Y}_{{l}_{2}+\frac{1}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)-{c}^{\frac{1}{2}}\frac{{\mu }_{n2}}{{a}_{2}}{Y}_{{l}_{2}+\frac{3}{2}}\left(\frac{{\mu }_{n2}}{{a}_{2}}c\right)\right]\left[G{\phi }_{0,0}^{3}\left(c,d\right)+H{\phi }_{0,1}^{3}\left(c,d\right)\right]\right\},\end{array}$ (24)

$\begin{array}{l}{M}_{3}=\frac{D}{\Delta }{\left(bc\right)}^{-\frac{1}{2}}{\left(\frac{\pi }{2}\right)}^{-2}{\phi }_{0,1}^{1}\left(b,b\right){\phi }_{0,1}^{1}\left(c,c\right)\left\{G{d}^{\frac{1}{2}}{J}_{{l}_{3}+\frac{1}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)\\ \text{}+H\left[\left({l}_{3}+1\right){d}^{-\frac{1}{2}}{J}_{{l}_{3}+\frac{1}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)-{d}^{\frac{1}{2}}\frac{{\mu }_{n3}}{{a}_{3}}{J}_{{l}_{3}+\frac{3}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)\right]\right\},\end{array}$ (25)

$\begin{array}{l}{N}_{3}=-\frac{D}{\Delta }{\left(bc\right)}^{-\frac{1}{2}}{\left(\frac{\pi }{2}\right)}^{-2}{\phi }_{0,1}^{1}\left(b,b\right){\phi }_{0,1}^{2}\left(c,c\right)\left\{G{d}^{\frac{1}{2}}{Y}_{{l}_{3}+\frac{1}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)\\ \text{}+H\left[\left({l}_{3}+1\right){d}^{-\frac{1}{2}}{Y}_{{l}_{3}+\frac{1}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)-{d}^{\frac{1}{2}}\frac{{\mu }_{n3}}{{a}_{3}}{Y}_{{l}_{3}+\frac{3}{2}}\left(\frac{{\mu }_{n3}}{{a}_{3}}d\right)\right]\right\}.\end{array}$ (26)

${y}_{1}={\Phi }_{1}\left(x\right),$ (27)

${\Phi }_{3}\left(x\right)=\frac{{\phi }_{0,0}^{3}\left(x,d\right)}{{\phi }_{1,0}^{3}\left(c,d\right)},$ (28)

${\Phi }_{3}\left(x\right)=\frac{{\phi }_{0,1}^{3}\left(x,d\right)}{{\phi }_{1,1}^{3}\left(c,d\right)},$ (29)

${\left[{y}_{1}\left(x\right)+F{y}_{1}{}^{\prime }\left(x\right)\right]|}_{x=a}=\frac{D}{E+\frac{1}{F+{\Phi }_{1}\left(a\right)}}.$ (30)

4. 相似构造法的求解步骤

5. 实例

$\left\{\begin{array}{l}{x}^{2}{y}_{1}{}^{\prime \text{​}\prime }+\left[{x}^{2}-2\right]{y}_{1}=0,\left(1\le x\le 2\right)\\ {x}^{2}{y}_{2}{}^{\prime \text{​}\prime }+\left[4{x}^{2}-6\right]{y}_{2}=0,\left(2\le x\le 3\right)\\ {x}^{2}{y}_{3}{}^{\prime \text{​}\prime }+\left[9{x}^{2}-12\right]{y}_{3}=0,\left(3\le x\le 4\right)\\ {\left[2{y}_{1}+7{y}_{1}{}^{\prime }\right]|}_{x=1}=2,\\ {{y}_{1}|}_{x=2}={{y}_{2}|}_{x=2},{{y}_{1}{}^{\prime }|}_{x=2}={2{y}_{2}{}^{\prime }|}_{x=2},\\ {{y}_{2}|}_{x=3}={{y}_{3}|}_{x=3},{{y}_{2}{}^{\prime }|}_{x=3}={2{y}_{3}{}^{\prime }|}_{x=3},\\ {\left[3{y}_{3}+4{y}_{3}{}^{\prime }\right]|}_{x=4}=0,\end{array}$ (31)

${\phi }_{0,0}^{1}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{x\xi }{\psi }_{\frac{3}{2},\frac{3}{2}}\left(x,\xi ,1\right),$

${\phi }_{0,1}^{1}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{\frac{x}{\xi }}\left[2{\psi }_{\frac{3}{2},\frac{3}{2}}\left(x,\xi ,1\right)-\xi {\psi }_{\frac{3}{2},\frac{5}{2}}\left(x,\xi ,1\right)\right],$

${\phi }_{1,0}^{1}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{\frac{\xi }{x}}\left[2{\psi }_{\frac{3}{2},\frac{3}{2}}\left(x,\xi ,1\right)-x{\psi }_{\frac{5}{2},\frac{3}{2}}\left(x,\xi ,1\right)\right],$

${\phi }_{1,1}^{1}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{\frac{1}{x\xi }}\left[4{\psi }_{\frac{3}{2},\frac{3}{2}}\left(x,\xi ,1\right)-2\xi {\psi }_{\frac{3}{2},\frac{5}{2}}\left(x,\xi ,1\right)-2x{\psi }_{\frac{5}{2},\frac{3}{2}}\left(x,\xi ,1\right)+x\xi {\psi }_{\frac{5}{2},\frac{5}{2}}\left(x,\xi ,1\right)\right].$

${\phi }_{0,0}^{2}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{x\xi }{\psi }_{\frac{5}{2},\frac{5}{2}}\left(x,\xi ,2\right),$

${\phi }_{0,1}^{2}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{\frac{x}{\xi }}\left[3{\psi }_{\frac{5}{2},\frac{5}{2}}\left(x,\xi ,2\right)-2\xi {\psi }_{\frac{5}{2},\frac{7}{2}}\left(x,\xi ,2\right)\right],$

${\phi }_{1,0}^{2}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{\frac{\xi }{x}}\left[3{\psi }_{\frac{5}{2},\frac{5}{2}}\left(x,\xi ,2\right)-2x{\psi }_{\frac{7}{2},\frac{5}{2}}\left(x,\xi ,2\right)\right],$

${\phi }_{1,1}^{2}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{\frac{1}{x\xi }}\left[9{\psi }_{\frac{5}{2},\frac{5}{2}}\left(x,\xi ,2\right)-6\xi {\psi }_{\frac{5}{2},\frac{7}{2}}\left(x,\xi ,2\right)-6x{\psi }_{\frac{7}{2},\frac{5}{2}}\left(x,\xi ,2\right)+4x\xi {\psi }_{\frac{7}{2},\frac{7}{2}}\left(x,\xi ,2\right)\right].$

${\phi }_{0,0}^{3}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{x\xi }{\psi }_{\frac{7}{2},\frac{7}{2}}\left(x,\xi ,3\right),$

${\phi }_{0,1}^{3}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{\frac{x}{\xi }}\left[4{\psi }_{\frac{7}{2},\frac{7}{2}}\left(x,\xi ,3\right)-3\xi {\psi }_{\frac{7}{2},\frac{9}{2}}\left(x,\xi ,3\right)\right],$

${\phi }_{1,0}^{3}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{\frac{\xi }{x}}\left[4{\psi }_{\frac{7}{2},\frac{7}{2}}\left(x,\xi ,3\right)-3x{\psi }_{\frac{9}{2},\frac{7}{2}}\left(x,\xi ,3\right)\right],$

${\phi }_{1,1}^{3}\left(x,\xi \right)=\frac{\pi }{2}\sqrt{\frac{1}{x\xi }}\left[16{\psi }_{\frac{7}{2},\frac{7}{2}}\left(x,\xi ,3\right)-12\xi {\psi }_{\frac{7}{2},\frac{9}{2}}\left(x,\xi ,3\right)-12x{\psi }_{\frac{9}{2},\frac{7}{2}}\left(x,\xi ,3\right)+9x\xi {\psi }_{\frac{9}{2},\frac{9}{2}}\left(x,\xi ,3\right)\right].$

${\Phi }_{1}\left(x\right)=\frac{2{\phi }_{0,0}^{1}\left(x,2\right)-{\Phi }_{2}\left(2\right){\phi }_{0,1}^{1}\left(x,2\right)}{2{\phi }_{1,0}^{1}\left(1,2\right)-{\Phi }_{2}\left(2\right){\phi }_{1,1}^{1}\left(1,2\right)},$

${\Phi }_{2}\left(x\right)=\frac{2{\phi }_{0,0}^{2}\left(x,3\right)-{\Phi }_{3}\left(3\right){\phi }_{0,1}^{2}\left(x,3\right)}{2{\phi }_{1,0}^{2}\left(2,3\right)-{\Phi }_{3}\left(3\right){\phi }_{1,1}^{2}\left(2,3\right)},$

${\Phi }_{3}\left(x\right)=\frac{3{\phi }_{0,0}^{3}\left(x,4\right)+4{\phi }_{0,1}^{3}\left(x,4\right)}{3{\phi }_{1,0}^{3}\left(3,4\right)+4{\phi }_{1,1}^{3}\left(3,4\right)}.$

${y}_{1}=2\frac{1}{2+\frac{1}{3+{\Phi }_{1}\left(1\right)}}\cdot \frac{1}{3+{\Phi }_{1}\left(1\right)}\cdot {\Phi }_{1}\left(x\right),$

${y}_{2}=2\frac{1}{2+\frac{1}{3+{\Phi }_{1}\left(1\right)}}\cdot \frac{1}{3+{\Phi }_{1}\left(1\right)}\cdot \frac{{\phi }_{0,1}^{1}\left(2,2\right)}{{\Phi }_{2}\left(2\right){\phi }_{1,1}^{1}\left(1,2\right)-2{\phi }_{1,0}^{1}\left(1,2\right)}\cdot {\Phi }_{2}\left(x\right),$

$\begin{array}{l}{y}_{3}=2\frac{1}{2+\frac{1}{3+{\Phi }_{1}\left(1\right)}}\cdot \frac{1}{3+{\Phi }_{1}\left(1\right)}\cdot \frac{{\phi }_{0,1}^{1}\left(2,2\right)}{{\Phi }_{2}\left(2\right){\phi }_{1,1}^{1}\left(1,2\right)-2{\phi }_{1,0}^{1}\left(1,2\right)}\\ \text{}\cdot \frac{{\phi }_{0,1}^{2}\left(3,3\right)}{{\Phi }_{3}\left(3\right){\phi }_{1,1}^{2}\left(2,3\right)-2{\phi }_{1,0}^{2}\left(2,3\right)}{\Phi }_{3}\left(x\right).\end{array}$

6. 结论

1) 三区复合型Riccati-Bessel方程边值问题的内区、中区、外区解都呈现连分数的形式且具有相似性，并且针对不同边界条件系数的设定，其解的相似结构仍然保持不变。

2) 对于三区复合型Riccati-Bessel方程边值问题，先利用定解方程的线性无关解构造的内、中、外区引解函数，再利用引解函数和外边界、衔接面条件的系数构造相似核函数，最后组装内边界系数、引解函数和相似核函数即可得到边值问题的解。

3) 利用相似构造法求解三区复合型Riccati-Bessel方程边值问题时，简化了求解过程，加快了求解速度，得到的解表达式简洁，更能清晰地反映各个参数对解的影响。然而，这一方法仍需要进一步深入研究和探索，以确定其在处理更复杂的复合型Riccati-Bessel方程边值问题以及边值问题为双边非齐次的问题时的有效性和适用性。

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