The Euler command represents Euler's operator: it returns zero when applied to an expression that is the divergence of another expression also called "current". When this divergence is also equal to zero this current is called conserved current.

Denoting the unknowns of a DE system by U, their derivatives by dU and the independent variables by X, Euler's operator is constructed as (d/dU + (-1)^n d/dX d/ddU), where n is the differential order of dU, d/dX is the differentiation operator such that dU = d/dx U and in Euler's operator there is one term of the form (-1)^n d/dX d/ddU for each dU in the DE system.

If DepVars is not given, Euler will consider all the differentiated unknown functions in PDESYS as unknowns of the problems. Specifying DepVars however permits not only restricting the unknowns in different ways but also specifying unknowns of the problems which do not appear differentiated in PDESYS.

When applying Euler's operator to a DE system constituted by N equations in M unknowns, there will be N x M expressions; that is: each of the M unknowns generates an operator of the form just described and each of these operators generates one expression when applied to each of the N equations of the system.

This set of expressions resulting from applying Euler's operator to a DE system, representing the conditions for the DE system to be the divergence of a current, is by default simplified with respect to its integrability conditions, then each expression in the set is simplified with respect to its size. This default behavior can be changed by providing one or more of the arguments integrabilityconditions = false, simplifier = none, or simplifier = any_desired_simplification.

To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

$\mathrm{with}\left(\mathrm{PDEtools}\right)\:$

$U≔\mathrm{diff\_table}\left(u\left(x\,t\right)\right)\:$

$\mathrm{declare}\left(U\left[\right]\right)$

$u\left(x\,t\right)\mathrm{will\; now\; be\; displayed\; as}u$

$\mathrm{pde}\left[1\right]≔U\left[t,t\right]+U\left[x,x\right]+U\left[x\right]U\left[\right]=0$

${\mathrm{pde}}_1≔u{u}_x+u_{t,t}+u_{x,x}=0$

$\mathrm{Euler}\left(\mathrm{pde}\left[1\right]\right)$

$\left\{0\right\}$

$\mathrm{ConservedCurrents}\left(\mathrm{pde}\left[1\right]\right)$

$\left[{\mathrm{\_J}}_x\left(x\,t\,u\,u_x\,u_t\right)={\mathrm{D}}_3\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)u_t+{\mathrm{D}}_2\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)+{\mathrm{f\_\_5}}_t+\frac{u^2}{2}+u_x\,{\mathrm{\_J}}_t\left(x\,t\,u\,u_x\,u_t\right)=-{\mathrm{f\_\_5}}_x-{\mathrm{D}}_3\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)u_x-{\mathrm{D}}_1\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)+\mathrm{f\_\_6}\left(x\right)+u_t\right],\left[{\mathrm{\_J}}_x\left(x\,t\,u\,u_x\,u_t\right)={\mathrm{D}}_3\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)u_t+{\mathrm{D}}_2\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)+{\mathrm{f\_\_5}}_t+\frac{u^{2}t}{2}+u_{x}t\,{\mathrm{\_J}}_t\left(x\,t\,u\,u_x\,u_t\right)=-{\mathrm{f\_\_5}}_x-{\mathrm{D}}_3\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)u_x-{\mathrm{D}}_1\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)+\mathrm{f\_\_6}\left(x\right)+t{u}_t-u\right]$

$\mathrm{IntegratingFactors}\left(\mathrm{pde}\left[1\right]\right)$

$\left[{\mathrm{\_\μ}}_1\left(x\,t\,u\,u_x\,u_t\right)=1\right],\left[{\mathrm{\_\μ}}_1\left(x\,t\,u\,u_x\,u_t\right)=t\right]$

$\mathrm{pde}\left[2\right]≔\mathrm{\Delta }\left(x\,t\,U\left[\right]\,U\left[x\right]\,U\left[t\right]\right)=0$

${\mathrm{pde}}_2≔\mathrm{\Delta }\left(x\,t\,u\,u_x\,u_t\right)=0$

$\mathrm{conditions\_for\_divergence}≔\mathrm{Euler}\left(\mathrm{pde}\left[2\right]\right)$

$\mathrm{conditions\_for\_divergence}≔\left[{\mathrm{\Delta }}_{u,u_x}=\frac{-{\mathrm{\Delta }}_{u,u_t}{u}_t-{\mathrm{\Delta }}_{t,u_t}-{\mathrm{\Delta }}_{x,u_x}+{\mathrm{\Delta }}_u}{u_x}\,{\mathrm{\Delta }}_{u_x,u_x}=0\,{\mathrm{\Delta }}_{u_t,u_x}=0\,{\mathrm{\Delta }}_{u_t,u_t}=0\right]$

$\mathrm{Divergence\_1PDE\_general\_form}≔\mathrm{pdsolve}\left(\mathrm{conditions\_for\_divergence}\right)$

$\mathrm{Divergence\_1PDE\_general\_form}≔\left\{\mathrm{\Delta }\left(x\,t\,u\,u_x\,u_t\right)={\mathrm{f\_\_3}}_u{u}_x+{\mathrm{f\_\_4}}_u{u}_t+{\mathrm{f\_\_3}}_x+{\mathrm{f\_\_4}}_t+\mathrm{f\_\_6}\left(x\,t\right)\right\}$

$\mathrm{answer}≔\mathrm{FromJet}\left(\mathrm{rhs}\left(\mathrm{Divergence\_1PDE\_general\_form}\left[1\right]\right)\,u\left(x\,t\right)\right)=0$

$\mathrm{answer}≔{\mathrm{D}}_3\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)u_x+{\mathrm{D}}_3\left(\mathrm{f\_\_4}\right)\left(x\,t\,u\right)u_t+{\mathrm{D}}_1\left(\mathrm{f\_\_3}\right)\left(x\,t\,u\right)+{\mathrm{D}}_2\left(\mathrm{f\_\_4}\right)\left(x\,t\,u\right)+\mathrm{f\_\_6}\left(x\,t\right)=0$

$\mathrm{Euler}\left(\mathrm{answer}\,u\left(x\,t\right)\right)$

$\left\{0\right\}$

$\mathrm{declare}\left(A\left(x\,t\right)\right)$

$A\left(x\,t\right)\mathrm{will\; now\; be\; displayed\; as}A$

$\mathrm{pde}\left[3\right]≔A\left[0\right]\left(x\,t\right)U\left[\right]+A\left[1\right]\left(x\,t\right)U\left[x\right]+A\left[2\right]\left(x\,t\right)U\left[t\right]+A\left[3\right]\left(x\,t\right)U\left[x,x\right]+A\left[4\right]\left(x\,t\right)U\left[x,t\right]+A\left[5\right]\left(x\,t\right)U\left[t,t\right]=0$

${\mathrm{pde}}_3≔A_{0}u+A_1{u}_x+A_2{u}_t+A_3{u}_{x,x}+A_4{u}_{t,x}+A_5{u}_{t,t}=0$

$\mathrm{Euler}\left(\mathrm{pde}\left[3\right]\,U\left[\right]\right)$

$\left\{A_0-{\left(A_2\right)}_t-{\left(A_1\right)}_x+{\left(A_5\right)}_{t,t}+{\left(A_4\right)}_{t,x}+{\left(A_3\right)}_{x,x}\right\}$

$\mathrm{condition}≔\mathrm{isolate}\left(\left[1\right]\,A\left[0\right]\left(x\,t\right)\right)$

$\mathrm{condition}≔A_0={\left(A_2\right)}_t+{\left(A_1\right)}_x-{\left(A_5\right)}_{t,t}-{\left(A_4\right)}_{t,x}-{\left(A_3\right)}_{x,x}$

$\mathrm{Divergence\_linear\_2PDE}≔\mathrm{subs}\left(\mathrm{condition}\,\mathrm{pde}\left[3\right]\right)$

$\mathrm{Divergence\_linear\_2PDE}≔\left({\left(A_2\right)}_t+{\left(A_1\right)}_x-{\left(A_5\right)}_{t,t}-{\left(A_4\right)}_{t,x}-{\left(A_3\right)}_{x,x}\right)u+A_1{u}_x+A_2{u}_t+A_3{u}_{x,x}+A_4{u}_{t,x}+A_5{u}_{t,t}=0$

$\mathrm{Euler}\left(\mathrm{Divergence\_linear\_2PDE}\,U\left[\right]\right)$

$\left\{0\right\}$