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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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\left\{0\right\}$

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

$\left\{0\right\}$

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

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

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

$\left\{0\right\}$

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