In this section we introduce the compact finite difference schemes and illustrate how we use them to solve nonlinear evolution partial differential equations. We consider nonlinear PDEs of the form
$$\frac{\mathrm{\partial}u}{\mathrm{\partial}t}=G(u,\frac{\mathrm{\partial}u}{\mathrm{\partial}x},\frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}{x}^{2}})$$(1)

defined on the the region
$$t\in [0,T],\phantom{\rule{1em}{0ex}}x\in [a,b]$$

with boundary condition
$$u(a,t)={u}_{a},\phantom{\rule{1em}{0ex}}u(b,t)={u}_{b},$$

In the derivation of the compact FD schemes we consider the function *u*(*x*, *t*) that depends on space variable *x* and temporal variable *t*. We consider one-dimensional uniform meshes on the regions [*a*, *b*] and [0, *T*], with nodes *x*_{i} (*i* = 1, 2, …, *N*_{x}) and *t*_{j} (*j* = 1, 2, …, *N*_{t}) respectively, where
$$a={x}_{1}<{x}_{2}<\cdots <{x}_{{N}_{\mathrm{x}}}=b$$(2)

and
$$0={t}_{1}<{t}_{2}<\cdots <{t}_{{N}_{\mathrm{x}}}=T$$(3)

The distance between any two successive nodes is a constant Δ *x* = *x*_{i} – *x*_{i–1} for the space variable and Δ *t* = *t*_{j}–*t*_{j–1} for the temporal variable. We first show how we approximate the derivatives with respect to *x* using the compact FD schemes. Sixth order approximations of the first and second derivatives with respect to *x*, at interior nodes can be obtained using the following schemes (see [6] for details)
$$\begin{array}{}{\displaystyle \frac{1}{3}{u}_{i-1,j}^{\prime}+{u}_{i,j}^{\prime}+\frac{1}{3}{u}_{i+1,j}^{\prime}=\frac{14}{9}\frac{{u}_{i+1,j}-{u}_{i-1,j}}{2\mathrm{\Delta}x}+\frac{1}{9}\frac{{u}_{i+2,j}-{y}_{i-2,j}}{4\mathrm{\Delta}x},}\end{array}$$(4)
$$\begin{array}{}{\displaystyle \frac{2}{11}{u}_{i-1,j}^{\u2033}+{u}_{i,j}^{\u2033}+\frac{2}{11}{u}_{i+1,j}^{\u2033}=\frac{12}{11}\frac{{u}_{i+1,j}-2{u}_{i,j}+{u}_{i-1,j}}{(\mathrm{\Delta}x{)}^{2}}+\frac{3}{11}\frac{{u}_{i+2,j}-2{u}_{i,j}+{u}_{i-2,j}}{4(\mathrm{\Delta}x{)}^{2}},}\end{array}$$(5)

where *u*_{i,j} = *u*(*x*_{i}, *t*_{j}) and the primes denote differentiation with respect to *x*. We apply the compact FD approximation for the first and second derivatives given by (4) and (5) respectively, at the interior nodes (*i* = 2, …, *N*_{x} − 1). Since we know boundary conditions at *i* = 1 and *i* = *N*_{x}, the compact FD schemes must be adjusted for the nodes near the boundary points. In order to maintain the order *O*(*h*^{6}) accuracy at the boundary points as in the interior points and to maintain the same tridiagonal format, we use the following one sided scheme at *i* = 2,
$$\begin{array}{}{u}_{2,j}^{\prime}+{\displaystyle \frac{1}{3}{u}_{3,j}^{\prime}=\frac{1}{\mathrm{\Delta}x}(-\frac{7}{45}{u}_{1,j}-\frac{17}{12}{u}_{2,j}+\frac{83}{36}{u}_{3,j}-\frac{11}{9}{u}_{4,j}+\frac{2}{3}{u}_{5,j}-\frac{37}{180}{u}_{6,j}+\frac{1}{36}{u}_{7,j}),}\end{array}$$(6)

and when *i* = *N*_{x} − 1 we use
$$\begin{array}{}{\displaystyle \frac{1}{3}{u}_{{N}_{x}-2,j}^{\prime}+{u}_{{N}_{x}-1,j}^{\prime}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& ={\displaystyle \frac{1}{\mathrm{\Delta}x}(\frac{7}{45}{u}_{{N}_{X},j}+\frac{17}{12}{u}_{{N}_{x}-1,j}-\frac{83}{36}{u}_{{N}_{x}-2,j}+\frac{11}{9}{u}_{{N}_{x}-3,j}-\frac{2}{3}{u}_{{N}_{x}-4,j}}\\ & +{\displaystyle \frac{37}{180}{u}_{{N}_{x}-5,j}+\frac{1}{36}{u}_{{N}_{x}-6,j}),}\end{array}$$(7)

Similarly, for the second derivatives, we use
$$\begin{array}{}{\displaystyle {u}_{2,j}^{\u2033}+\frac{2}{11}{u}_{3,j}^{\u2033}=\frac{1}{(\mathrm{\Delta}x{)}^{2}}(\frac{31}{45}{u}_{1,j}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle -\frac{19}{110}{u}_{2,j}-\frac{339}{110}{u}_{3,j}+\frac{1933}{396}{u}_{4,j}-\frac{40}{11}{u}_{5,j}}\\ & {\displaystyle +\frac{96}{55}{u}_{6,j}-\frac{479}{990}{u}_{7,j}+\frac{13}{220}{u}_{8,j}),}\end{array}$$(8)

at *i* = 2 and
$$\begin{array}{}{\displaystyle \frac{2}{11}{u}_{{N}_{x}-2,j}^{\u2033}+{u}_{{N}_{x}-1,j}^{\u2033}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle =\frac{1}{(\mathrm{\Delta}x{)}^{2}}(\frac{31}{45}{u}_{{N}_{x},j}+\frac{19}{110}{u}_{{N}_{x}-1,j}+\frac{339}{110}{u}_{{N}_{x}-2,j}+\frac{1933}{396}{u}_{{N}_{x}-3,j}}\\ & -{\displaystyle \frac{40}{11}{u}_{{N}_{x}-4,j}\frac{96}{55}{u}_{{N}_{x}-5,j}+\frac{479}{990}{u}_{{N}_{x}-6,j}+\frac{13}{220}{u}_{{N}_{x}-7,j}),}\end{array}$$(9)

at *i* = *N*_{x} − 1. Using the above equations, the equations for approximating the first and second order derivatives can be expressed as
$$\begin{array}{}{\mathbf{A}}_{x}{\mathbf{U}}_{\cdot ,j}^{\prime}={\mathbf{B}}_{x}{\mathbf{U}}_{\cdot ,j}+{\mathbf{K}}_{x},\end{array}$$(10)
$$\begin{array}{}{\mathbf{A}}_{xx}{\mathbf{U}}_{\cdot ,j}^{\u2033}={\mathbf{B}}_{xx}{\mathbf{U}}_{\cdot ,j}+{\mathbf{K}}_{xx},\end{array}$$(11)

where
$$\begin{array}{c}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{A}}_{x}={\left[\begin{array}{cccc}& 1& \frac{1}{3}& \\ & \frac{1}{3}& 1& \frac{1}{3}& \\ & & \frac{1}{3}& 1& \frac{1}{3}\\ & & & \ddots & \ddots & \ddots & \\ & & & & \frac{1}{3}& 1& \frac{1}{3}& \\ & & & & & \frac{1}{3}& 1& \end{array}\right]}_{({N}_{X}-2)\times ({N}_{x}-2)},{\mathbf{K}}_{x}=\frac{1}{\mathrm{\Delta}x}{\left[\begin{array}{c}-\frac{7}{45}{y}_{1}\\ -\frac{{y}_{1}}{36}\\ 0\\ \vdots \\ \vdots \\ 0\\ -\frac{{y}_{N}}{36}\\ \frac{7}{45}{y}_{N}\end{array}\right]}_{({N}_{X}-2)\times 1}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{B}}_{x}=\frac{1}{\mathrm{\Delta}x}{\left[\begin{array}{ccccccc}& -\frac{17}{12}& \frac{83}{36}& -\frac{11}{9}& \frac{2}{3}& -\frac{37}{180}& \frac{1}{36}\\ & -\frac{7}{9}& 0& \frac{7}{9}& \frac{1}{36}\\ & \frac{-1}{36}& -\frac{7}{9}& 0& \frac{7}{9}& \frac{1}{36}\\ & & \ddots & \ddots & \ddots & \ddots & \ddots \\ & & & \frac{-1}{36}& -\frac{7}{9}& 0& \frac{7}{9}& \frac{1}{36}\\ & & & & \frac{-1}{36}& -\frac{7}{9}& 0& \frac{7}{9}\\ & & -\frac{1}{36}& \frac{37}{180}& -\frac{2}{3}& \frac{11}{9}& \frac{83}{36}& \frac{17}{12}\end{array}\right]}_{({N}_{x}-2)\times ({N}_{x}-2)}\\ \phantom{\rule{thinmathspace}{0ex}}{\mathbf{A}}_{xx}={\left[\begin{array}{cccc}& 1& \frac{2}{11}& \\ & \frac{2}{11}& 1& \frac{2}{11}& \\ & & \frac{2}{11}& 1& \frac{2}{11}\\ & & & \ddots & \ddots & \ddots & \\ & & & & \frac{2}{11}& 1& \frac{2}{11}& \\ & & & & & \frac{2}{11}& 1& \end{array}\right]}_{({N}_{x}-2)\times ({N}_{x}-2)},{K}_{xx}=\frac{1}{(\mathrm{\Delta}x{)}^{2}}{\left[\begin{array}{c}\frac{31}{45}{y}_{1}\\ \frac{3{y}_{1}}{44}\\ 0\\ \vdots \\ \vdots \\ 0\\ \frac{3{y}_{N}}{44}\\ \frac{31}{45}{y}_{N}\end{array}\right]}_{({N}_{x}-2)\times 1}\\ \phantom{\rule{thinmathspace}{0ex}}{\mathbf{B}}_{xx}=\frac{1}{(\mathrm{\Delta}x{)}^{2}}{\left[\begin{array}{cccccccc}& -\frac{19}{110}& -\frac{339}{110}& \frac{1933}{396}& -\frac{40}{11}& \frac{96}{55}& -\frac{479}{990}& \frac{13}{220}\\ & -\frac{12}{11}& -\frac{51}{22}& \frac{12}{11}& \frac{3}{44}\\ & \frac{3}{44}& \frac{12}{11}& -\frac{51}{22}& \frac{12}{11}& \frac{3}{44}\\ & & \ddots & \ddots & \ddots & \ddots & \ddots \\ & & & & \frac{3}{44}& \frac{12}{11}& -\frac{51}{22}& \frac{12}{11}& \frac{3}{44}\\ & & & & & \frac{3}{44}& \frac{12}{11}& -\frac{51}{22}& \frac{12}{11}\\ & & -\frac{13}{220}& -\frac{479}{990}& \frac{96}{55}& -\frac{40}{11}& \frac{1933}{396}& -\frac{339}{110}& \frac{19}{110}\end{array}\right]}_{({N}_{x}-2)\times ({N}_{x}-2)}\\ {\mathbf{U}}_{\cdot ,j}^{\prime}=[{u}_{2,j}^{\prime},\phantom{\rule{thinmathspace}{0ex}}{u}_{3,j}^{\prime},\phantom{\rule{thinmathspace}{0ex}}\dots ,\phantom{\rule{thinmathspace}{0ex}}{u}_{{N}_{x}-2,j}^{\prime},\phantom{\rule{thinmathspace}{0ex}}{u}_{{N}_{x}-1,j}^{\prime}{]}^{T},\phantom{\rule{1em}{0ex}}{\mathbf{U}}_{\cdot ,j}^{\u2033}=[{u}_{2,j}^{\u2033},\phantom{\rule{thinmathspace}{0ex}}{u}_{3,j}^{\u2033},\phantom{\rule{thinmathspace}{0ex}}\dots ,\phantom{\rule{thinmathspace}{0ex}}{u}_{{N}_{x}-2,j}^{\u2033},\phantom{\rule{thinmathspace}{0ex}}{u}_{{N}_{x}-1,j}^{\u2033}{]}^{T}\end{array}$$

From equations (10) and (11), *U*′ and *U*″ can be expressed as;
$$\begin{array}{}{\mathbf{U}}_{\cdot ,j}^{\prime}={\mathbf{E}}_{x}{\mathbf{U}}_{\cdot ,j}+{\mathbf{H}}_{x}\end{array}$$(12)
$$\begin{array}{}{\mathbf{U}}_{\cdot ,j}^{\u2033}={\mathbf{E}}_{xx}{\mathbf{U}}_{\cdot ,j}+{\mathbf{H}}_{xx}\end{array}$$(13)

where
$$\begin{array}{}{\mathbf{E}}_{x}={\mathbf{A}}_{x}^{-1}{\mathbf{B}}_{x},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{E}}_{xx}={\mathbf{A}}_{xx}^{-1}{\mathbf{B}}_{xx},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{H}}_{x}={\mathbf{A}}_{x}^{-1}{\mathbf{K}}_{x},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{H}}_{xx}={\mathbf{A}}_{xx}^{-1}{\mathbf{K}}_{xx}\end{array}$$(14)

Next we show how we use the compact FD schemes to approximate the time derivatives. Approximation of the first derivative at interior points is given by
$$\begin{array}{}{\displaystyle \frac{1}{3}{\dot{u}}_{i,j-1}+{\dot{u}}_{i,j}+\frac{1}{3}{\dot{u}}_{i,j+1}=\frac{14}{9}\frac{{u}_{i,j+1}-{u}_{i,j-1}}{2\mathrm{\Delta}t}+\frac{1}{9}\frac{{u}_{i,j+2}-{y}_{i,j-2}}{4\mathrm{\Delta}t},}\end{array}$$(15)

where the dots denote differentiation with respect to time and Δ *t* is the distance between successive nodes. We adjust the schemes for *i* = 1, 2, *N*_{t} − 1 and *N*_{t} with the following one sided schemes respectively.
$$\begin{array}{}{\displaystyle {\dot{u}}_{i,1}+\frac{1}{3}{\dot{u}}_{i,2}=\frac{1}{\mathrm{\Delta}t}(-\frac{451}{180}{u}_{i,1}+\frac{1003}{180}{u}_{i,2}-\frac{20}{3}{u}_{i,3}+\frac{55}{9}{u}_{i,4}-\frac{125}{36}{u}_{i,5}+\frac{67}{60}{u}_{i,6}-\frac{7}{45}{u}_{i,7}),}\end{array}$$(16)
$$\begin{array}{}{\displaystyle \frac{1}{3}{\dot{u}}_{i,1}+{\dot{u}}_{i,2}+\frac{1}{3}{\dot{u}}_{i,3}=\frac{1}{\mathrm{\Delta}t}(-\frac{35}{36}{u}_{i,1}+\frac{7}{12}{u}_{i,2}-\frac{7}{36}{u}_{i,3}+{u}_{i,4}-\frac{7}{12}{u}_{i,5}+\frac{7}{36}{u}_{i,6}-\frac{1}{36}{u}_{i,7}),}\end{array}$$(17)
$$\begin{array}{}{\displaystyle \frac{1}{3}{\dot{u}}_{i,{N}_{t}}-1+{\dot{u}}_{i,{N}_{t}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle =\frac{1}{\mathrm{\Delta}t}(\frac{35}{36}{u}_{i,{N}_{t}}-\frac{7}{12}{u}_{i,{N}_{t}-1}+\frac{7}{36}{u}_{i,{N}_{t}-2}-{u}_{i,{N}_{t}-3}+\frac{7}{12}{u}_{i,{N}_{t}-4}}\\ & -{\displaystyle \frac{7}{36}{u}_{i,{N}_{t}-5}+\frac{1}{36}{u}_{i,{N}_{t}-6}),}\end{array}$$(18)
$$\begin{array}{}{\displaystyle \frac{1}{3}{\dot{u}}_{i,{N}_{t}-2}+{\dot{u}}_{i,{N}_{t}-1}+\frac{1}{3}{\dot{u}}_{i,{N}_{t}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& ={\displaystyle \frac{1}{\mathrm{\Delta}t}(\frac{451}{180}{u}_{i,{N}_{t}}-\frac{1003}{180}{u}_{i,{N}_{t}-1}+\frac{20}{3}{u}_{i,{N}_{t}-2}-\frac{55}{9}{u}_{i,{N}_{t}-3}}\\ & -{\displaystyle \frac{125}{36}{u}_{i,{N}_{t}-4}+\frac{67}{60}{u}_{i,{N}_{t}-5}-\frac{7}{45}{u}_{i,{N}_{t}-6}),}\end{array}$$(19)

The equation for approximating the first time derivative is obtained by combining equations (15)-(19) and is given by
$$\begin{array}{}{\mathbf{A}}_{t}{\dot{\mathbf{U}}}_{i,}.={\mathbf{B}}_{t}{\mathbf{U}}_{i,}.\end{array}$$(20)

where **A**_{t} = **A**_{x} and
$$\begin{array}{}{\mathbf{B}}_{t}={\displaystyle \frac{1}{\mathrm{\Delta}t}}{\left[\begin{array}{cccccccccc}-\frac{451}{180}& \frac{1003}{180}& -\frac{20}{3}& \frac{55}{9}& -\frac{125}{36}& \frac{67}{60}& -\frac{7}{45}& & & \\ -\frac{35}{36}& \frac{7}{12}& -\frac{7}{36}& 1& -\frac{7}{12}& \frac{7}{36}& -\frac{1}{36}& & & \\ {\displaystyle -\frac{1}{36}}& {\displaystyle -\frac{7}{9}}& 0& {\displaystyle \frac{7}{9}}& {\displaystyle \frac{1}{36}}\\ & \ddots & \ddots & \ddots & \ddots & \ddots & & \\ & & & {\displaystyle -\frac{1}{36}}& {\displaystyle -\frac{7}{9}}& 0& {\displaystyle \frac{7}{9}}& {\displaystyle \frac{1}{36}}\\ & \frac{1}{36}& -\frac{7}{36}& \frac{7}{12}& -1& \frac{7}{36}& -\frac{7}{12}& \frac{35}{36}\\ & \frac{7}{45}& -\frac{67}{60}& \frac{125}{36}& -\frac{55}{9}& \frac{20}{3}& -\frac{1003}{180}& \frac{451}{180}\end{array}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\right]}_{{N}_{t}\times {N}_{t}}\end{array}$$

From equation (20), *U̇* can be expressed as
$$\begin{array}{}{\displaystyle {\dot{\mathbf{U}}}_{i,\cdot}={\mathbf{E}}_{t}{\mathbf{U}}_{i,\cdot}=\sum _{k=1}^{{N}_{t}}{e}_{j,k}u({x}_{i},\phantom{\rule{thinmathspace}{0ex}}{t}_{k}),\phantom{\rule{2em}{0ex}}i=2,3,\dots ,{N}_{x}-1,\phantom{\rule{1em}{0ex}}j=1,2,\dots ,{N}_{t}}\end{array}$$(21)

where ${\mathbf{E}}_{t}={\mathbf{A}}_{t}^{-1}{\mathbf{B}}_{t}$ and *e*_{jk} are the elements of **E**_{t} and **U**_{i,⋅} = [*u*_{i,1}, *u*_{i,2}, …, *u*_{i,Nt}].

To solve the PDE (1) we start by linearizing it by using the quasilinearization method which was proposed by Bellman and Kalaba [12]. It is convenient to split the function *G* in (1) into its linear and nonlinear components and rewrite the governing equation in the form,
$$\begin{array}{}L[u,{u}^{\prime},{u}^{\u2033}]+N[u,{u}^{\prime},{u}^{\u2033}]-\dot{u}=0,\end{array}$$(22)

where the dot and primes denote the time and space derivatives, respectively. *L* is a linear operator and *N* is a non-linear operator. If we assume that the difference *u*_{r+1} − *u*_{r} and all its space derivatives is small, then we can approximate the non-linear operator *N* using the linear terms of the Taylor series and hence
$$\begin{array}{}N[u,{\displaystyle {u}^{\prime},{u}^{\u2033}]\approx N[{u}_{r},{u}_{r}^{\prime},{u}_{r}^{\u2033}]+\sum _{k=0}^{2}\frac{\mathrm{\partial}N}{\mathrm{\partial}{u}^{(k)}}({u}_{r+1}^{(k)}-{u}_{r}^{(k)})}\end{array}$$(23)

where *r* and *r* + 1 denote previous and current iterations respectively.

Equation (23) can be expressed as
$$\begin{array}{}N[u,{\displaystyle {u}^{\prime},{u}^{\u2033}]\approx N[{u}_{r},{u}_{r}^{\prime},{u}_{r}^{\u2033}]+\sum _{k=0}^{2}{\varphi}_{k,r}[{u}_{r},{u}_{r}^{\prime},{u}_{r}^{\u2033}]{u}_{r+1}^{(k)}-\sum _{k=0}^{2}{\varphi}_{k,r}[{u}_{r},{u}_{r}^{\prime},{u}_{r}^{\u2033}]{u}_{r}^{(k)}}\end{array}$$(24)

where
$$\begin{array}{}{\displaystyle {\varphi}_{k,r}[{u}_{r},{u}_{r}^{\prime},{u}_{r}^{\u2033}]=\frac{\mathrm{\partial}N}{\mathrm{\partial}{u}^{(k)}}[{u}_{r},{u}_{r}^{\prime},{u}_{r}^{\u2033}].}\end{array}$$(25)

Substituting equation (24) into equation (22), we get
$$\begin{array}{}L[{u}_{r+1},{\displaystyle {u}_{r+1}^{\prime},{u}_{r+1}^{\u2033}]+\sum _{k=0}^{2}{\varphi}_{k,r}{u}_{r+1}^{(k)}-{\dot{u}}_{r+1}={R}_{r}[{u}_{r},{u}_{r}^{\prime},{u}_{r}^{\u2033}]}\end{array}$$(26)

where
$$\begin{array}{}{\displaystyle {R}_{r}[{u}_{r},{u}_{r}^{\prime},{u}_{r}^{\u2033}]=\sum _{k=0}^{2}{\varphi}_{k,r}{u}_{r}^{(k)}-N[{u}_{r}2,{u}_{r}^{\prime},{u}_{r}^{\u2033}].}\end{array}$$

Substituting (21) into (26) we get
$$\begin{array}{}L[{\displaystyle {\mathbf{U}}_{r+1,j},{\mathbf{U}}_{r+1,j}^{\prime},{\mathbf{U}}_{r+1,j}^{\u2033}]+\sum _{k=0}^{2}{\mathbf{\Phi}}_{k,r}{\mathbf{U}}_{r+1,j}^{(k)}-2\sum _{k=0}^{{N}_{t}}{e}_{jk}{\mathbf{U}}_{r+1,k}={R}_{r}[{\mathbf{U}}_{r,j},{\mathbf{U}}_{r,j}^{\prime},{\mathbf{U}}_{r,j}^{\u2033}]}\end{array}$$(27)

for *j* = 1, 2, 3, …, *N*_{t}, where
$$\begin{array}{}{\mathbf{\Phi}}_{k,r}=\left[\begin{array}{}{\varphi}_{k,r}({x}_{2},{t}_{j})& & & \\ & \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\varphi}_{k,r}({x}_{1},{t}_{j})& & \\ & & \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\ddots & \\ & & & \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\varphi}_{k,r}({x}_{{N}_{x}-1},\phantom{\rule{thinmathspace}{0ex}}{t}_{j})\end{array}\right].\end{array}$$(28)

and
$$\begin{array}{}{\mathbf{U}}_{r+1,j}=[{u}_{r+1},j({x}_{2}),{u}_{r+1},j({x}_{3}),\dots ,{u}_{r+1},j({x}_{{N}_{x}-1})]\end{array}$$(29)

Since the initial condition is known, then we express equation (27) as
$$\begin{array}{}L[{\displaystyle {\mathbf{U}}_{r+1,j},{\mathbf{U}}_{r+1,j}^{\prime},{\mathbf{U}}_{r+1,j}^{\u2033}]+\sum _{k=0}^{2}{\mathrm{\Phi}}_{k,r}{\mathbf{U}}_{r+1,j}^{(k)}-\sum _{k=1}^{{N}_{t}}{e}_{j,k}{\mathbf{U}}_{r+1,k}={\mathbf{R}}_{j}}\end{array}$$(30)

where
$$\begin{array}{}{\mathbf{R}}_{j}={R}_{r}[{\mathbf{U}}_{r,j},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{U}}_{r,j}^{\prime},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{U}}_{r,j}^{\u2033}]+{e}_{i,1}{\mathbf{U}}_{1}-L[{\mathbf{H}}_{x},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{H}}_{xx}]-{\mathbf{\Phi}}_{1,r}{\mathbf{H}}_{x}-{\mathbf{\Phi}}_{2,r}{\mathbf{H}}_{xx},\phantom{\rule{1em}{0ex}}j=2,3,\dots ,{N}_{t}.\end{array}$$(31)

Equation (30) can be expressed as the following (*N*_{t} − 1)(*N*_{x} − 1)×(*N*_{t} − 1)(*N*_{x} − 1) matrix system
$$\begin{array}{}\left[\begin{array}{cccc}{X}_{2,2}& {X}_{2,3}& \cdots & {X}_{2,{N}_{t}}\\ {X}_{3,2}& {X}_{3,3}& \cdots & {X}_{3,{N}_{t}}\\ \vdots & \vdots & \ddots & \vdots \\ {X}_{{N}_{t},2}& {X}_{{N}_{t},3}& \cdots & {X}_{{N}_{t},{N}_{t}}\end{array}\right]\left[\begin{array}{c}{\mathbf{U}}_{2}\\ {\mathbf{U}}_{3}\\ \vdots \\ {\mathbf{U}}_{{N}_{t}}\end{array}\right]=\left[\begin{array}{c}{\mathbf{R}}_{2}\\ {\mathbf{R}}_{3}\\ \vdots \\ {\mathbf{R}}_{{N}_{t}}\end{array}\right],\end{array}$$(32)

where
$$\begin{array}{}{X}_{i,i}=L[\mathbf{I},{\mathbf{E}}_{x},{\mathbf{E}}_{xx}]+{\mathbf{\Phi}}_{0,r}+{\mathbf{\Phi}}_{1,r}{\mathbf{E}}_{x}+{\mathbf{\Phi}}_{2,r}{\mathbf{E}}_{xx}-{e}_{i,i}\mathbf{I}\end{array}$$(33)
$$\begin{array}{}{X}_{i,j}=-{e}_{i,j}\mathbf{I},\phantom{\rule{1em}{0ex}}\text{when\hspace{0.17em}}i\ne j,\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i,j=2,3,\dots ,{N}_{t}\end{array}$$(34)

and **I** is the identity matrix of size (*N*_{x} − 1) × (*N*_{x} − 1). Solving equation (28) gives *u*(*x*_{i}, *t*_{j}).

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