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A solution to a partial differential equation that factors into functions of separate variables, for example $u(x,t)=X(x)T(t)$. For the heat equation $u_t=D{u}_{xx}$, substitution yields $T^{\prime }/(DT)=X^{\prime \prime }/X$, so both sides are a common constant wherever division is valid. The resulting ordinary differential equations must be solved together with the original boundary conditions. Individual separated solutions generate more general solutions by superposition when the equation and boundary data are linear.

A similarity ansatz assumes a profile invariant under a prescribed rescaling, commonly in the displayed power-law form. Substitution into a partial differential equation fixes compatible exponents and can reduce it to an ordinary differential equation for $F$. Boundary conditions and conserved quantities must also be compatible with the scaling; the ansatz alone does not establish existence of a solution.

The weak form of $(L^{\ast }-\lambda )u=f$ includes the same scalar shift as the differential equation. The unshifted identity $B^{\ast }[u,v]=(f,v)$ instead represents $L^{\ast }u=f$ and cannot imply the shifted equation for nonzero $\lambda$ unless $u=0$. On a smooth bounded domain choose any nonzero compactly supported smooth $u$ and take $L=-\Delta +1$, $f=Lu$ for a direct counterexample. For constant drift coefficients, integration by parts reverses the drift sign in $L^{\ast }$ without changing this spectral term.

A stationary solution of an evolution differential equation is independent of time. Substituting ${\partial }_{t}u=0$ leaves the spatial boundary value problem that determines its profile. For an ordinary differential equation this is an equilibrium; for a partial differential equation it may vary in space. A stationary profile need not have zero flux, although zero flux is forced in a one-dimensional conserved system with reflecting endpoints.

A time-stepping method approximates the evolution of an ordinary differential equation by successive updates at discrete times. It can be applied after a spatial finite element method has converted a time-dependent partial differential equation into an ordinary differential equation system. The implicit time-stepping method is one class of such methods.

For a partial differential equation $F(x,u,Du,D^{2}u)=0$ whose operator is nonincreasing in the Hessian variable, a continuous viscosity subsolution satisfies $F(x_0,u(x_0),D\varphi (x_0),D^2\varphi (x_0))\le 0$ whenever a smooth test function $\varphi$ touches it from above. A viscosity supersolution satisfies the reverse inequality for tests touching from below. A viscosity solution satisfies both conditions. For $F=-\mathrm{tr}{D}^{2}u$, these conditions are respectively $\Delta \varphi \ge 0$ and $\Delta \varphi \le 0$. They make sense even when $u$ has no classical second derivatives.

A travelling wave of the viscous Burgers equation can connect constant states $u_L>u_R$ through a smooth decreasing layer. Require $u(x,t)=U(z)$, $z=x-st$, $U(-\infty )=u_L$, $U(+\infty )=u_R$, and $U^{\prime }\to 0$ at both ends. Substitution and one integration giveThe speed follows by evaluating the integration constant at both end states. It is precisely the Rankine-Hugoniot condition for conservation law flux $f(u)=u^2/2$. Choosing the center $U(0)=s$ gives the displayed hyperbolic tangent profile. Its characteristic width scales as $\nu /(u_L-u_R)$; a smaller positive $\nu$ makes a thinner smooth layer.

For an original worked example in dimensionless quantities, take $u_L=2$, $u_R=0$ and $s=1$:This is an exact classical solution on $\mathbb{R}$ with its own smooth initial condition $u(x,0)=1-\tanh (x/(2\nu ))$ and limiting states $2,0$. It is a family of moving profiles, not the finite-$\nu$ solution starting from an exact discontinuous step. For that different initial condition, the Cole-Hopf transformation supplies the evolution. At $\nu =1/2$, this front starts from the same $1-\tanh x$ profile as the Burgers characteristic breaking example: positive diffusion maintains a smooth moving layer while the inviscid profile breaks at $t=1$.

To check the partial differential equation explicitly, set $w=(x-t)/(2\nu )$ and $S=1-{\tanh }^{2}w$. ThenThe independent Rankine-Hugoniot condition check gives $s=(0-2)/(0-2)=1$, using $f(2)=2$ and $f(0)=0$. At fixed $x\ne t$, the vanishing viscosity approximation tends to the entropy shock $2$ for $x<t$ and $0$ for $x>t$. At the center the smooth profiles stay at $1$; changing the value on the discontinuity does not change the limiting weak solution.

For $u_L<u_R$, the same integrated ordinary differential equation has $U^{\prime }<0$ between the two states when $\nu >0$, so it cannot supply an increasing connecting front. The inviscid admissible expansion is the rarefaction wave of the Burgers Riemann problem with positive flux.

For $x{f}_x+\beta y{f}_y=\gamma f$, the curves in the method of characteristics are $x=x_0{e}^s$, $y=y_0{e}^{\beta s}$ and $f=f_0{e}^{\gamma s}$. On either component $x\ne 0$, the invariant is $y|x{|}^{-\beta }$, giving $f=|x{|}^{\gamma }F(y|x{|}^{-\beta })$. A coordinate change that includes $y/x$ may be singular on an axis, so smooth continuation there must be checked against the original partial differential equation.