Using operator method in solving nonlinear differential equations

Zenonas Navickas

Department of Applied Mathematics, Kaunas University of Technology

zenavi@fmf.ktu.lt

Contributed talk

The latest developments in computer hardware as well as in software facilities stipulate appearance of new interesting computational algorithms. These algorithms distinguish themselves for higher efficiency, serve the purpose and, in most cases, lead to more accurate results. But it must be admitted that these algorithms are associated with increased computer capacities and time expenditure.

Some newly developed algorithms are based on the application of operator calculus. From the practical point of view, such algorithms are especially convenient in representing solutions of differential equations by means of algebraic expressions of standard operators.

Some observations. Let $\mathbf{D}_v$, $\mathbf{D}_s$, ... and $\mathbf{L}_x$ be linear differential and integral operators with respect to the variables $v$, $s$, ... and $x$ (i.e. $\mathbf{D}_vv^n=nv^{n-1}$, ..., $${\mathbf{L}_xx^n=\frac{x^{n+1}}{n+1}}$$, $n=0,1,2,\dots$). Then the following operators can be formed -- a generalized differential operator $\mathbf{D}_{vst}\stackrel{\mathrm{def}}{=}S\mathbf{D}_v+Q\mathbf{D}_s+K\mathbf{D}_t$ and a linear multiplicative operator $${\mathbf{G}\stackrel{\mathrm{def}}{=}\sum^{+\infty}_{k=0}\left(\mathbf{L}_x\mathbf{D}_{vst}\right)^k}$$, where $S=S(x,v,s,t)$, ... are arbitrary polynomials (here $${(\mathbf{L}_x\mathbf{D}_{vst})^0=\mathbf{L}_x^0\mathbf{D}^0_{vst}=\mathbf{1}}$$ is a unitary operator). In this case, the following equality holds true

$$\mathbf{G}f(v,s,t)=f(\mathbf{G}v,\mathbf{G}s,\mathbf{G}t),$$

for all series $${\sum^{+\infty}_{k,l,r=0}a_{klr}v^ks^lt^r}$$, $a_{klr}\in \mathbf{R}$.

The operator $\mathbf{G}$ is used in representing solutions of both the ordinary and the partial differential equations. For instance, the solution of the nonlinear ordinary differential equation $y_{xx}''=P(x,y,y_x')$, $y=y(x,s,t,v)$, $y(v,s,t,v)=s$, $y_x'(x,s,t,v)\vert _{x=v}=t$, is written in the form of the operator-series

$$y(x,s,t,v)=\sum^{+\infty}_{k=0}\left(\left(\mathbf{D}_v+t\ma... ...{D}_s+P(v,s,t)\mathbf{D}_t\right)^ks\right)\frac{(x-v)^k}{k!}.$$

Let $u_x'-u_y'=T(u)$ be a nonlinear partial differential equation, where $u=u(x,y)$, $u(0,y)=\varphi_0(y)$ and $\varphi_0(y)$ is given. Then

$$u=u_0(x,\varphi(x,y)),$$

where $${u_0(x,s)=\sum^{+\infty}_{k=0}\left((T(s)\mathbf{D}_s)^ks\right)\frac{x^k}{k!}}$$; $${\varphi(x,y)=\sum^{+\infty}_{k=0}(\mathbf{L}_x\mathbf{D}_y)^k\varphi_0(y)}$$.

Post - script. The above operator calculus based methodology might be used in further developing of computational methods such as Runge-Kutta's method and others.