<Text-field style="Heading 1" layout="Heading 1">Assignments to partial derivatives</Text-field>

coordinates([t,x], [u,v], 8);

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<Text-field style="Heading 2" layout="Heading 2">Example 1</Text-field>

put('pd(U,u_x^2)' = -u_x*pd(U,u));

pd(U,u_x^2);

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pd(U,u_x^3);

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clear(pds);

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pd(U,u_x^3);

LUkjcGRHNiI2JEkiVUdGJCokSSR1X3hHRiQiIiQ=

<Text-field style="Heading 2" layout="Heading 2">Example 2</Text-field>

Resolving Cauchy\342\200\223Riemann conditions

put(pd(U,u) = pd(V,v), pd(V,u) = -pd(U,v));

produces the Laplace equation

pd(U,u^2);

i.e.

pd(U,u^2)+pd(U,v^2)=0;

clear(pds):

<Text-field style="Heading 2" layout="Heading 2">Cross-derivatives and compatibility conditions</Text-field>

Substitutions can contradict one another, generating chaotic and session-dependent results.

For instance, the above Cauchy\342\200\223Riemann conditions can be alternatively resolved as

put( 'pd(V,v)' = pd(U,u), 'pd(V,u)' = -pd(U,v));

but then the input

pd(V,u*v);

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may produce different results depending on whether it resulted from the \357\254\201rst or the second substitution.

cc();

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put('pd(U,u^2)' = -pd(U,v^2));

cc();

PCI=

<Text-field style="Heading 2" layout="Heading 2">Varodering</Text-field>

clear(pds):

dependence(U(t, x, u, u_x,u_xx), V(x,u,u_x));

unknowns(U,V);

S := [U, pd(U,u),pd(U,u^2), pd(U,u*x), pd(U,x), pd(U,t), pd(U,u_x), pd(U,u_xx), V, pd(V,u), pd(V,x*u), pd(V,x), pd(V,u_x)];

E := V-pd(U,u_x);

Varordering(degree,function,reverse);

sort(S, `Vars/<<`);

resolve(E);

Varordering(function,degree,reverse);

sort(S, `Vars/<<`);

resolve(E);

<Text-field style="Heading 2" layout="Heading 2">Trick (two or more unknowns)</Text-field>

dependence(U(t, x,u,v,u_x), V(t, x,u,v,u_x));

unknowns(U,V);

S := TheExpression;

Varordering(degree,function,reverse):

run(S);

Z := clear(pds);

Varordering(function,degree,reverse):

run(Z);