Preface
This section presents some applications of power series method in numerical approximation of solutions to systems of ordinary differential equations.
Return to computing page for the second course APMA0340
Return to Mathematica tutorial for the first course APMA0330
Return to Mathematica tutorial for the second course APMA0340
Return to the main page for the first course APMA0330
Return to the main page for the second course APMA0340
Return to Part VI of the course APMA0340
Introduction to Linear Algebra
Glossary
Klein--Gordon equation
The Klein--Gordon and sine-Gordon equations are a two nonlinear hyperbolic partial differential equations that model problems in classical and quantum mechanics, solitons, and condensed matter physics. Let us consider the Klein--Gordon equation
\[
u_{tt} - y_{xx} + b\, u + g(u) = f(x,t) ,
\]
and sine-Gordon equation
\[
u_{tt} - y_{xx} + \alpha\, \sin (u) = f(x,t) ,
\]
subject to the initial conditions
\[
u(x,0) = f_0 (x), \qquad u_t (x,0) = f_1 (x) .
\]
Example: Consider the linear Klein--Gordon equation
\[
\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} +u ,
\]
subject to the initial conditions
\[
u(x,0) = 1 + \sin x, \qquad u_t (x,0) =0.
\]
■
Example: Consider the following nonlinear Klein--Gordon equation:
\[
\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}
- \left( u(x,t) \right)^2 + x^2 t^2 ,
\]
subject to the initial conditions
\[
u(x,0) =0, \qquad u_t (x,0) = x.
\]
We seek solution in the form:
\[
u(x,t) = \sum_{j\ge 0} \sum_{j\ge 0} a_{i,j} x^i t^j .
\]
Then
\begin{align*}
\frac{\partial^2 u}{\partial t^2} &= \sum_{j\ge 0} \sum_{j\ge 0} \left( j+1 \right) \left( j+2 \right) a_{i,j+2} x^i t^j ,
\\
\frac{\partial^2 u}{\partial x^2} &=\sum_{j\ge 0} \sum_{j\ge 0} \left( i+1 \right) \left( i+2 \right) a_{i+2,j} x^i t^j ,
\\
u^2 (x,t) &= \sum_{j\ge 0} \sum_{j\ge 0} A_{i,j} x^i t^j ,
\end{align*}
where
\[
A_{i,j} = \frac{1}{i!\,j!} \left. \frac{\partial^{i+j}}{\partial x^i \,\partial t^j} \, u^2 (x,t) \right\vert_{x=0, t=0} , \qquad A_{0,0} = a_{0,0}^2 .
\]
In particular,
\begin{align*}
A_{0,1} &= 2\,a_{0,0} a_{0,1} ,
\\
A_{0,2} &= 2\,a_{0,0} a_{0,2} + a_{0,1}^2 ,
\\
\vdots & \quad \vdots
\\
A_{1,0} &= 2\,a_{0,0} a_{1,0} ,
\\
A_{1,1} &= 2\,a_{0,0} a_{1,1} + 2\,a_{0,1} a_{1,0}
\\
\vdots & \quad \vdots
\\
A_{2,0} &= 2\,a_{0,0} a_{2,0} + a_{1,0}^2 ,
\end{align*}
and so on. Then, by substituting these formulas into the given equation, we obtain
\[
\sum_{j\ge 0} \sum_{j\ge 0} \left( j+1 \right)\left( j+2 \right) a_{i,j+2} x^i t^j =
\sum_{j\ge 0} \sum_{j\ge 0} \left( i+1 \right) \left( i+2 \right) a_{i+2,j} x^i t^j - \sum_{j\ge 0} \sum_{j\ge 0} A_{i,j} x^i t^j + x^2 t^2 ,
\]
and by equating like powers, we get
\[
\left( j+1 \right)\left( j+2 \right) a_{i,j+2} =
\left( i+1 \right) \left( i+2 \right) a_{i+2,j} - A_{i,j} +
\delta_{i-2, j-2} ,
\]
where
\[
\delta_{\nu , \mu} = \begin{cases}
1 , & \ \mbox{ if } \nu = \mu =0 ,
\\
0, & \ \mbox{ otherwise}
\end{cases}
\]
is the Kronecker's delta. Now we take into account the initial conditions
\begin{align*}
u(x,0) &= \sum_{i\ge 0} a_{i,0} x^i =0 \qquad \Longrightarrow \qquad
a_{i,0} =0;
\\
u_t (x,0) &= \sum_{i\ge 0} a_{i,1} x^i = x \qquad \Longrightarrow \qquad
a_{i,1} = \delta_{i-1} .
\end{align*}
This allows us to determine explicitly the values of coefficients and obtain
\[
u(x,t) = xt .
\]
■
- Yousif, M.A. and Mahmood, B.A., Approximate solutions for solving theKlein–Gordon and sine-Gordon equations, Journal of the Association of Arab Universities forBasic and Applied Sciences, 2017, Vol. 22, pp. 83--90.
Return to Mathematica page
Return to the main page (APMA0340)
Return to the Part 1 Matrix Algebra
Return to the Part 2 Linear Systems of Ordinary Differential Equations
Return to the Part 3 Non-linear Systems of Ordinary Differential Equations
Return to the Part 4 Numerical Methods
Return to the Part 5 Fourier Series
Return to the Part 6 Partial Differential Equations
Return to the Part 7 Special Functions