Given a system $\mathrm{\Delta }=0$ consisting of N equations ${\mathrm{pde}}_n$, $n=1..N$, where the independent variables are $x_1,x_2,\mathrm{...}=X$, and the dependent variables are $u_1,u_2,\mathrm{...}=U$, with $\mathrm{dU}$ denoting the set of partial derivatives of $U$, the generalized integrating factors are expressions ${\mathrm{\mu }}_{\mathrm{\alpha },n}\left(X\,U\,\mathrm{dU}\right)$ such that $\mathrm{\Sigma }{\mathrm{\mu }}_{\mathrm{\alpha },n}{\mathrm{pde}}_n=\mathrm{Divergence}\left(J_{\mathrm{\alpha }}\right)$ = 0, so $J_{\mathrm{\alpha }}$ is a conserved current. These generalized integrating factors, also called characteristic functions of conserved currents (see reference [1]), coincide with the traditional integrating factors when there is only one independent variable, so that $\mathrm{\Delta }$ is a system of ODEs.

The command IntegratingFactors computes these generalized integrating factors. The command IntegratingFactorTest verifies the result for correctness.

Given the system $\mathrm{\Delta }=0$ the output of IntegratingFactors is as a sequence of $\mathrm{\alpha }$ lists, each one containing N ${\mathrm{\mu }}_{\mathrm{\alpha },n}$, where $n=1..N$, satisfying $\mathrm{\Sigma }{\mathrm{\mu }}_{\mathrm{\alpha },n}{\mathrm{pde}}_n=\mathrm{Divergence}\left(J_{\mathrm{\alpha }}\right)$ = 0.

The ${\mathrm{\mu }}_{\mathrm{\alpha },n}\left(X\,U\,\mathrm{dU}\right)$ are computed constructing the PDE system they satisfy by applying Euler's operator to $\mathrm{\Sigma }{\mathrm{\mu }}_{\mathrm{\alpha },n}{\mathrm{pde}}_n=0$ , then solving this system for the ${\mathrm{\mu }}_{\mathrm{\alpha },n}$ using pdsolve.

By default, the integrating factors are searched as functions depending on the derivatives of the unknowns of the system (specified as DepVars or automatically detected) up to the order d-1, where d is the highest order of derivatives entering PDESYS. This default can be changed by optionally passing the argument order = m, where m is a nonnegative integer.

By default, the conserved currents are searched as functions with no pre-specified form, just with the dependency explained in the previous paragraph. This default can be changed with the option typeofconservedcurrent = ... where the right-hand-side can be polynomial or functionfield, respectively indicating a conserved current of polynomial type, or of a functionfield type with the meaning explained in FunctionField.

By default, the functionality of ${\mathrm{\mu }}_{\mathrm{\alpha },n}\left(X\,U\,\mathrm{dU}\right)$, entering the left-hand-sides of each element in the returned lists, is displayed, the output is presented in functional notation instead of jet notation and is simplified with respect to its size. The PDE system solved to compute the ${\mathrm{\mu }}_{\mathrm{\alpha },n}\left(X\,U\,\mathrm{dU}\right)$ is also split, when that is possible, before being tackled. All these defaults can be changed by passing the optional arguments displayfunctionality = ..., jetnotation = ..., simplifier = ..., split = false.

It is also possible to directly specify the functionality expected for the ${\mathrm{\mu }}_{\mathrm{\alpha },n}$ using _mu = .... See the examples for a demonstration of the use of this parameter.

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.

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

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

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

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

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

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

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

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

$\mathrm{map}\left(\mathrm{IntegratingFactorTest}\,\left[\mathrm{\mu }\left[\mathrm{\alpha }\right]\right]\,\mathrm{pde}\left[1\right]\right)$

$\left[\left\{0\right\}\,\left\{0\right\}\right]$

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

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

$\mathrm{map}\left(\mathrm{ConservedCurrentTest}\,\left[J\left[\mathrm{\alpha }\right]\right]\,\mathrm{pde}\left[1\right]\right)$

$\left[\left\{0\right\}\,\left\{0\right\}\right]$

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

${\mathrm{pde}}_2≔u{u}_x+u_{t,x}+u_{x,x,x}=0$

$\mathrm{IntegratingFactors}\left(\mathrm{pde}\left[2\right]\right)$

$\left[{\mathrm{\_\μ}}_1\left(x\,t\,u\,u_x\,u_t\,u_{x,x}\,u_{t,x}\,u_{t,t}\right)=\mathrm{f\_\_1}\left(t\,\frac{u^2}{2}+u_t+u_{x,x}\right)\right]$

$\mathrm{IntegratingFactors}\left(\mathrm{pde}\left[2\right]\,\mathrm{order}=1\right)$

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

$\mathrm{ConservedCurrents}\left(\mathrm{pde}\left[2\right]\,\mathrm{order}=1\right)$

$\left[{\mathrm{\_J}}_x\left(x\,t\,u\,u_x\,u_t\right)={\mathrm{D}}_3\left(\mathrm{f\_\_1}\right)\left(x\,t\,u\right)u_t+{\mathrm{D}}_2\left(\mathrm{f\_\_1}\right)\left(x\,t\,u\right)+{\mathrm{f\_\_3}}_t\,{\mathrm{\_J}}_t\left(x\,t\,u\,u_x\,u_t\right)=-{\mathrm{D}}_3\left(\mathrm{f\_\_1}\right)\left(x\,t\,u\right)u_x-{\mathrm{D}}_1\left(\mathrm{f\_\_1}\right)\left(x\,t\,u\right)-{\mathrm{f\_\_3}}_x+\mathrm{f\_\_4}\left(x\right)\right]$

$\mathrm{IntegratingFactors}\left(\mathrm{pde}\left[2\right]\,\mathrm{\_\μ}=f\left(U\left[x\right]\,U\left[t\right]\right)\right)$

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

$\mathrm{IntegratingFactors}\left(\mathrm{pde}\left[2\right]\,\mathrm{\_\μ}=f\left(U\left[x,x\right]\,U\left[t\right]\,U\left[\right]\right)\right)$

$\left[{\mathrm{\_\μ}}_1\left(u_{x,x}\,u_t\,u\right)=\mathrm{f\_\_1}\left(u^2+2u_t+2u_{x,x}\right)\right]$

$\mathrm{IntegratingFactors}\left(\mathrm{pde}\left[2\right]\,\mathrm{type}=\mathrm{polynomial}\,\mathrm{split}=\mathrm{false}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}type\text{'}\; against\; keyword\; \text{'}typeofintegratingfactor\text{'}}$

$\left[{\mathrm{\_\μ}}_1\left(x\,t\,u\,u_x\,u_t\,u_{x,x}\,u_{t,x}\,u_{t,t}\right)=\left(u^2+2u_t+2u_{x,x}\right)\mathrm{c\_\_3}+\mathrm{c\_\_4}{t}^2+\mathrm{c\_\_2}t+\mathrm{c\_\_1}\right]$

$\mathrm{IntegratingFactorTest}\left(\,\mathrm{pde}\left[2\right]\right)$

$\left\{0\right\}$

$\mathrm{Euler}\left(\mathrm{pde}\left[2\right]\right)$

$\left\{0\right\}$

$J\left[\mathrm{\alpha }\right]≔\mathrm{ConservedCurrents}\left(\mathrm{pde}\left[2\right]\,\mathrm{\_J}=f\left(U\left[x,x\right]\,U\left[t\right]\,U\left[\right]\right)\right)$

$J_{\mathrm{\alpha }}≔\left[{\mathrm{\_J}}_x\left(u_{x,x}\,u_t\,u\right)=\mathrm{f\_\_1}\left(u^2+2u_t+2u_{x,x}\right)\,{\mathrm{\_J}}_t\left(u_{x,x}\,u_t\,u\right)=1\right]$

$\mathrm{ConservedCurrentTest}\left(J\left[\mathrm{\alpha }\right]\,\mathrm{pde}\left[2\right]\right)$

$\left\{0\right\}$

[1] Olver, P.J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. Springer-Verlag, 1993.