kill(all)$

solve pde (part 6), use laplace transform. 'version 1.0'
peter.vlasschaert@gmail.com,11/12/2018

load 'packages'
------------------------

load(orthopoly)$
load(lrats)$
load(facexp)$
load(odes)$

load("C:/math/laplaceInv.mac");

assume(s>0);

pde : diff(u(x,t),x) + diff(u(x,t),t,1) = x, t > 0
           =========================================
           domain : x >0 , t > 0

BC :   u(0,t)  =0
IC   :   u(x,0) =0

eq:diff(u(x,t),x) + diff(u(x,t),t,1) = x;
bc :[u(0,t)  =0];
ic : [u(x,0) =0];

bc[1];

solve : pde with bc,ic
--------------------------------

1e) laplace transformation 'eq'
=========================

p1:laplace(eq,t,s);

p1a:laplace(u(x,t),t,s) =v;

2e) subst ic into 'p1'
=================

p2:subst([ic[1],p1a],p1);

3e) solve ode2 , 'p2'
================

p3:ode2(p2,v,x);

find : %c
-----------------------------------------
BC:   u(0,t)  =0    x=0  => v=0
-----------------------------------------

p3a:subst([x=0,v=0],p3);

p3b:solve(p3a,%c);

p3c:subst(p3b,p3);

4e) inverse laplace transform 'p3d'
=============================

p4:laplaceInv(rhs(p3c),s,t);

p4a:u(x,t) = p4;

5e) check the solution
===================

p5:subst([p4a],eq);

p5a:ev(p5,nouns);

p5b:combine(p5a);

I can't check the solution :easy way.