One may consider the following FBIVP:

$$\begin{array}{}{\displaystyle {D}_{{0}_{+}}^{\alpha}x(t)+\lambda {e}^{x(t)}=0,\phantom{\rule{thinmathspace}{0ex}}1<\alpha \le 2,\phantom{\rule{thinmathspace}{0ex}}0<t<1,}\\ x(0)={x}_{0},\phantom{\rule{thinmathspace}{0ex}}{x}^{\prime}(0)={x}_{0}^{\prime},\end{array}$$(4)

By BCM, one may have

$$\begin{array}{}{\displaystyle {x}_{n}(t)\simeq \sum _{i=0}^{n}{c}_{i}{B}_{i,n}(t),\phantom{\rule{thinmathspace}{0ex}}0\le t\le 1,}\end{array}$$(5)

by Eqs. (4) and (5), one may have

$$\begin{array}{}{\displaystyle {D}_{{0}_{+}}^{\alpha}\left(\sum _{i=0}^{n}{c}_{i}{B}_{i,n}(t)\right)=-\lambda e{}^{\left(\sum _{i=0}^{n}{c}_{i}{B}_{i,n}(t)\right)}}\end{array}$$(6)

Hence

$$\begin{array}{}{\displaystyle R(x,{c}_{0},{c}_{1},\dots ,{c}_{n})=\sum _{i=0}^{n}{c}_{i}{D}_{{0}_{+}}^{\alpha}{B}_{i,n}(x)+\lambda {e}^{\left(\sum _{i=0}^{n}{c}_{i}{B}_{i,n}(t)\right)}}\end{array}$$

Suppose

$$\begin{array}{}{\displaystyle S(x,{c}_{0},{c}_{1},\dots ,{c}_{n})=\underset{0}{\overset{1}{\int}}R(x,{c}_{0},{c}_{1},\dots ,n{)}^{2}{w}_{1}(x)dx,}\end{array}$$

where *w*_{1}(*x*) = 1,

$$\begin{array}{}{\displaystyle S(x,{c}_{0},{c}_{1},\dots ,{c}_{n})=}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}={\displaystyle \underset{0}{\overset{1}{\int}}{\left(\sum _{i=0}^{n}{c}_{i}{D}^{\alpha}{B}_{i,n}(x)+\lambda {e}^{\left(\sum _{i=0}^{n}{c}_{i}{B}_{i,n}(t)\right)}\right)}^{2}dx,}\end{array}$$

then, one may have

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}S}{\mathrm{\partial}{c}_{i}}=0,\phantom{\rule{thinmathspace}{0ex}}0\le i\le n,}\end{array}$$

and therefore

$$\begin{array}{}{\displaystyle \underset{0}{\overset{1}{\int}}\left(\sum _{i=0}^{n}{c}_{i}{D}^{\alpha}{B}_{i,n}(x)+\lambda {e}^{\left(\sum _{i=0}^{n}{c}_{i}{B}_{i,n}(t)\right)}\right)}\\ \times \left({D}^{\alpha}{B}_{i,n}(x)+\lambda {e}^{\left(\sum _{i=0}^{n}{c}_{i}{B}_{i,n}(t)\right)}\left({B}_{i,n}(x)\right)\right)dx=0,\end{array}$$(7)

by Eq. (7), one can obtain a system of *n*+1 linear equations with *n*+1 unknown coefficients *c*_{i}. Also by utilizing many subroutine algorithm for solving this linear equations, one can find the unknown coefficients *c*_{i}, *i* = 0, 1, …, *n*.

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