As mentioned on the pdsolve/numeric page, before numerical solution of time based PDE systems containing higher than first order derivatives, the system is converted to an equivalent first order system for solution.

The standard method of doing this is to introduce new variables that take the place of the derivatives that are higher than first order in time, and by defining these new variables through equations that are first order in time.

The primary reason for this page is to explain that though this approach is straightforward to automate, it is not always the best solution. Other solutions, however, require reformulation of the input PDE or system, including any initial or boundary conditions, so often this is best done manually.  See the second example below.

From the examples below, it appears as though the choice of lowering the maximal differential order in $x$ is the best strategy for the first case, while lowering the error introduced in the boundary conditions is the best strategy for the second case.

This behavior can vary based on any aspect of the problem, the PDE, the initial conditions, or the boundary conditions.

In addition, in some cases the lowering of the differential order may be difficult due to mixed derivative terms or difficult boundary and initial conditions.

$\mathrm{diff}\left(u\left(x\,t\right)\,t\,t\right)=\mathrm{diff}\left(u\left(x\,t\right)\,x\,x\right)+\mathrm{diff}\left(u\left(x\,t\right)\,x\,t\right)$

$\frac{{\partial }^2}{\partial t^2}u\left(x\,t\right)=\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)+\frac{{\partial }^2}{\partial t\partial x}u\left(x\,t\right)$

$v\left(x\,t\right)=\mathrm{diff}\left(u\left(x\,t\right)\,t\right)$

$v\left(x\,t\right)=\frac{\partial }{\partial t}u\left(x\,t\right)$

$\mathrm{diff}\left(u\left(x\,t\right)\,t\right)=v\left(x\,t\right),\mathrm{diff}\left(v\left(x\,t\right)\,t\right)=\mathrm{diff}\left(u\left(x\,t\right)\,x\,x\right)+\mathrm{diff}\left(v\left(x\,t\right)\,x\right)$

$\frac{\partial }{\partial t}u\left(x\,t\right)=v\left(x\,t\right),\frac{\partial }{\partial t}v\left(x\,t\right)=\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)+\frac{\partial }{\partial x}v\left(x\,t\right)$

$\mathrm{PDE}≔\mathrm{diff}\left(u\left(x\,t\right)\,t\,t\right)=-\frac{1}{10}\mathrm{diff}\left(u\left(x\,t\right)\,x\,x\,x\,x\right)$

$\mathrm{PDE}≔\frac{{\partial }^2}{\partial t^2}u\left(x\,t\right)=-\frac{\frac{{\partial }^4}{\partial x^4}u\left(x\,t\right)}{10}$

$\mathrm{Init}≔u\left(x\,0\right)=\mathrm{uv}\left(x\right),\mathrm{D}\left[2\right]\left(u\right)\left(x\,0\right)=0$

$\mathrm{Init}≔u\left(x\,0\right)=\mathrm{uv}\left(x\right),{\mathrm{D}}_2\left(u\right)\left(x\,0\right)=0$

$\mathrm{Bdry}≔u\left(0\,t\right)=0,\mathrm{D}\left[1\right]\left(u\right)\left(0\,t\right)=0,u\left(1\,t\right)=0,\mathrm{D}\left[1\right]\left(u\right)\left(1\,t\right)=0$

$\mathrm{Bdry}≔u\left(0\,t\right)=0,{\mathrm{D}}_1\left(u\right)\left(0\,t\right)=0,u\left(1\,t\right)=0,{\mathrm{D}}_1\left(u\right)\left(1\,t\right)=0$

$\mathrm{PDE1}≔\mathrm{diff}\left(u\left(x\,t\right)\,t\right)=v\left(x\,t\right),\mathrm{diff}\left(v\left(x\,t\right)\,t\right)=-\frac{1}{10}\mathrm{diff}\left(u\left(x\,t\right)\,x\,x\,x\,x\right)$

$\mathrm{PDE1}≔\frac{\partial }{\partial t}u\left(x\,t\right)=v\left(x\,t\right),\frac{\partial }{\partial t}v\left(x\,t\right)=-\frac{\frac{{\partial }^4}{\partial x^4}u\left(x\,t\right)}{10}$

$\mathrm{Init1}≔u\left(x\,0\right)=\mathrm{uv}\left(x\right),v\left(x\,0\right)=0$

$\mathrm{Init1}≔u\left(x\,0\right)=\mathrm{uv}\left(x\right),v\left(x\,0\right)=0$

$\mathrm{Bdry1}≔u\left(0\,t\right)=0,\mathrm{D}\left[1\right]\left(u\right)\left(0\,t\right)=0,u\left(1\,t\right)=0,\mathrm{D}\left[1\right]\left(u\right)\left(1\,t\right)=0$

$\mathrm{Bdry1}≔u\left(0\,t\right)=0,{\mathrm{D}}_1\left(u\right)\left(0\,t\right)=0,u\left(1\,t\right)=0,{\mathrm{D}}_1\left(u\right)\left(1\,t\right)=0$

$\mathrm{PDE2}≔\mathrm{diff}\left(v\left(x\,t\right)\,t\right)=\frac{3}{10}\mathrm{diff}\left(u\left(x\,t\right)\,x\right),\mathrm{diff}\left(u\left(x\,t\right)\,t\right)=-\frac{1}{3}\mathrm{diff}\left(v\left(x\,t\right)\,x\,x\,x\right)$

$\mathrm{PDE2}≔\frac{\partial }{\partial t}v\left(x\,t\right)=\frac{3\frac{\partial }{\partial x}u\left(x\,t\right)}{10},\frac{\partial }{\partial t}u\left(x\,t\right)=-\frac{\frac{{\partial }^3}{\partial x^3}v\left(x\,t\right)}{3}$

$\mathrm{Init2}≔u\left(x\,0\right)=\mathrm{uv}\left(x\right),v\left(x\,0\right)=0$

$\mathrm{Init2}≔u\left(x\,0\right)=\mathrm{uv}\left(x\right),v\left(x\,0\right)=0$

$\mathrm{Bdry2}≔u\left(0\,t\right)=0,v\left(0\,t\right)=0,u\left(1\,t\right)=0,v\left(1\,t\right)=0$

$\mathrm{Bdry2}≔u\left(0\,t\right)=0,v\left(0\,t\right)=0,u\left(1\,t\right)=0,v\left(1\,t\right)=0$

$\mathrm{PDE3}≔\mathrm{diff}\left(v\left(x\,t\right)\,t\right)=\frac{3}{10}\mathrm{diff}\left(u\left(x\,t\right)\,x\,x\right),\mathrm{diff}\left(u\left(x\,t\right)\,t\right)=-\frac{1}{3}\mathrm{diff}\left(v\left(x\,t\right)\,x\,x\right)$

$\mathrm{PDE3}≔\frac{\partial }{\partial t}v\left(x\,t\right)=\frac{3\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)}{10},\frac{\partial }{\partial t}u\left(x\,t\right)=-\frac{\frac{{\partial }^2}{\partial x^2}v\left(x\,t\right)}{3}$

$\mathrm{Init3}≔u\left(x\,0\right)=\mathrm{uv}\left(x\right),v\left(x\,0\right)=0$

$\mathrm{Init3}≔u\left(x\,0\right)=\mathrm{uv}\left(x\right),v\left(x\,0\right)=0$

$\mathrm{Bdry3}≔u\left(0\,t\right)=0,\mathrm{D}\left[1\right]\left(u\right)\left(0\,t\right)=0,u\left(1\,t\right)=0,\mathrm{D}\left[1\right]\left(u\right)\left(1\,t\right)=0$

$\mathrm{Bdry3}≔u\left(0\,t\right)=0,{\mathrm{D}}_1\left(u\right)\left(0\,t\right)=0,u\left(1\,t\right)=0,{\mathrm{D}}_1\left(u\right)\left(1\,t\right)=0$

$\mathrm{apxsol}≔0.504413\left(\cosh \left(4.73x\right)-\cos \left(4.73x\right)\right)\cos \left(7.07505t\right)+0.495587\left(\sin \left(4.73x\right)-\sinh \left(4.73x\right)\right)\cos \left(7.07505t\right)$

$\mathrm{apxsol}≔0.504413\left(\cosh \left(4.73x\right)-\cos \left(4.73x\right)\right)\cos \left(7.07505t\right)+0.495587\left(\sin \left(4.73x\right)-\sinh \left(4.73x\right)\right)\cos \left(7.07505t\right)$

$\mathrm{uv}\left(x\right)≔\mathrm{eval}\left(\mathrm{apxsol}\,t=0\right)$

$\mathrm{uv}\left(x\right)≔0.504413\cosh \left(4.73x\right)-0.504413\cos \left(4.73x\right)+0.495587\sin \left(4.73x\right)-0.495587\sinh \left(4.73x\right)$

$\mathrm{pds1}≔\mathrm{pdsolve}\left(\left\{\mathrm{PDE1}\right\}\,\left\{\mathrm{Bdry1}\,\mathrm{Init1}\right\}\,\mathrm{numeric}\,\mathrm{timestep}=\frac{1}{32}\,\mathrm{spacestep}=\frac{1}{64}\right)$

$\mathrm{pds1}≔\mathbf{module}\left(\right)\mathbf{export}\mathrm{plot}\,\mathrm{plot3d}\,\mathrm{animate}\,\mathrm{value}\,\mathrm{settings}\;...\mathbf{end\; module}$

$\mathrm{pds2}≔\mathrm{pdsolve}\left(\left\{\mathrm{PDE2}\right\}\,\left\{\mathrm{Bdry2}\,\mathrm{Init2}\right\}\,\mathrm{numeric}\,\mathrm{timestep}=\frac{1}{32}\,\mathrm{spacestep}=\frac{1}{64}\right)$

$\mathrm{pds2}≔\mathbf{module}\left(\right)\mathbf{export}\mathrm{plot}\,\mathrm{plot3d}\,\mathrm{animate}\,\mathrm{value}\,\mathrm{settings}\;...\mathbf{end\; module}$

$\mathrm{pds3}≔\mathrm{pdsolve}\left(\left\{\mathrm{PDE3}\right\}\,\left\{\mathrm{Bdry3}\,\mathrm{Init3}\right\}\,\mathrm{numeric}\,\mathrm{timestep}=\frac{1}{32}\,\mathrm{spacestep}=\frac{1}{64}\right)$

$\mathrm{pds3}≔\mathbf{module}\left(\right)\mathbf{export}\mathrm{plot}\,\mathrm{plot3d}\,\mathrm{animate}\,\mathrm{value}\,\mathrm{settings}\;...\mathbf{end\; module}$

$\mathrm{pl1}≔\mathrm{pds1}:-\mathrm{plot}\left(\left[u\left(x\,t\right)-\mathrm{apxsol}\,\mathrm{color}=\mathrm{red}\right]\,t=5\right)\:$$\mathrm{pl2}≔\mathrm{pds2}:-\mathrm{plot}\left(\left[u\left(x\,t\right)-\mathrm{apxsol}\,\mathrm{color}=\mathrm{blue}\right]\,t=5\right)\:$$\mathrm{pl3}≔\mathrm{pds3}:-\mathrm{plot}\left(\left[u\left(x\,t\right)-\mathrm{apxsol}\,\mathrm{color}=\mathrm{green}\right]\,t=5\right)\:$$\mathrm{plots}\left[\mathrm{display}\right]\left(\left\{\mathrm{pl1}\,\mathrm{pl2}\,\mathrm{pl3}\right\}\right)$

$\mathrm{apxsol}≔0.5000168\left(\cosh \left(10.99561x\right)-\cos \left(10.99561x\right)\right)\cos \left(38.23301t\right)+0.5\left(\sin \left(10.99561x\right)-\sinh \left(10.99561x\right)\right)\cos \left(38.23301t\right)$

$\mathrm{apxsol}≔0.5000168\left(\cosh \left(10.99561x\right)-\cos \left(10.99561x\right)\right)\cos \left(38.23301t\right)+0.5\left(\sin \left(10.99561x\right)-\sinh \left(10.99561x\right)\right)\cos \left(38.23301t\right)$

$\mathrm{uv}\left(x\right)≔\mathrm{eval}\left(\mathrm{apxsol}\,t=0\right)$

$\mathrm{uv}\left(x\right)≔0.5000168\cosh \left(10.99561x\right)-0.5000168\cos \left(10.99561x\right)+0.5\sin \left(10.99561x\right)-0.5\sinh \left(10.99561x\right)$

$\mathrm{pds1}≔\mathrm{pdsolve}\left(\left\{\mathrm{PDE1}\right\}\,\left\{\mathrm{Bdry1}\,\mathrm{Init1}\right\}\,\mathrm{numeric}\,\mathrm{timestep}=\frac{1}{64}\,\mathrm{spacestep}=\frac{1}{64}\right)$

$\mathrm{pds1}≔\mathbf{module}\left(\right)\mathbf{export}\mathrm{plot}\,\mathrm{plot3d}\,\mathrm{animate}\,\mathrm{value}\,\mathrm{settings}\;...\mathbf{end\; module}$

$\mathrm{pds2}≔\mathrm{pdsolve}\left(\left\{\mathrm{PDE2}\right\}\,\left\{\mathrm{Bdry2}\,\mathrm{Init2}\right\}\,\mathrm{numeric}\,\mathrm{timestep}=\frac{1}{64}\,\mathrm{spacestep}=\frac{1}{64}\right)$

$\mathrm{pds2}≔\mathbf{module}\left(\right)\mathbf{export}\mathrm{plot}\,\mathrm{plot3d}\,\mathrm{animate}\,\mathrm{value}\,\mathrm{settings}\;...\mathbf{end\; module}$

$\mathrm{pds3}≔\mathrm{pdsolve}\left(\left\{\mathrm{PDE3}\right\}\,\left\{\mathrm{Bdry3}\,\mathrm{Init3}\right\}\,\mathrm{numeric}\,\mathrm{timestep}=\frac{1}{64}\,\mathrm{spacestep}=\frac{1}{64}\right)$

$\mathrm{pds3}≔\mathbf{module}\left(\right)\mathbf{export}\mathrm{plot}\,\mathrm{plot3d}\,\mathrm{animate}\,\mathrm{value}\,\mathrm{settings}\;...\mathbf{end\; module}$

$\mathrm{pl1}≔\mathrm{pds1}:-\mathrm{plot}\left(\left[u\left(x\,t\right)-\mathrm{apxsol}\,\mathrm{color}=\mathrm{red}\right]\,t=5\right)\:$$\mathrm{pl2}≔\mathrm{pds2}:-\mathrm{plot}\left(\left[u\left(x\,t\right)-\mathrm{apxsol}\,\mathrm{color}=\mathrm{blue}\right]\,t=5\right)\:$$\mathrm{pl3}≔\mathrm{pds3}:-\mathrm{plot}\left(\left[u\left(x\,t\right)-\mathrm{apxsol}\,\mathrm{color}=\mathrm{green}\right]\,t=5\right)\:$$\mathrm{plots}\left[\mathrm{display}\right]\left(\left\{\mathrm{pl1}\,\mathrm{pl2}\,\mathrm{pl3}\right\}\right)$