Convert the given 2nd order nonlinear PDE into a finite set of ODES Using Galerkin method D[u(x)] = -(x,t) + u"(x,t) + qu(xt)=0 (1) a2u here, (x,t)= at2 and u"(x,t)= axi 02u 2 Boundary conditions are given as u(0,t)=0 and u(L,t) = 0(2). Initial conditions are u(x,0) = u(x,0) = 0(3) = A trial function is (x, t) = {41)-4;(x) = a (t) 0.(x) +az(t). 02(x) +az(t)..(x) (4) Here, aj(t) are unknown parameters, not constants. 0;(x) are shape functions meeting the given boundary conditions: TEX = UN sin (5) (j = 1, 2, 3, ...) u(x, t) = : +4 + + 43 (x, t) = .. : +.. 02 +s. Da '(xt) = a, 01 + a2 0 +a, da " (xt) = a1.0" +22.0%+a, 0 (6) Dj sin MIX 0;=j, COS (7) j22 JTEX 0;' = II = Jei visin (') = - and 50:- o; dx = So for i =] (1 for i =) (8) 8 Weight function w(x) must be 0(x). Convert the given 2nd order nonlinear PDE into a finite set of ODES Using Galerkin method D[u(x)] = -(x,t) + u"(x,t) + qu(xt)=0 (1) a2u here, (x,t)= at2 and u"(x,t)= axi 02u 2 Boundary conditions are given as u(0,t)=0 and u(L,t) = 0(2). Initial conditions are u(x,0) = u(x,0) = 0(3) = A trial function is (x, t) = {41)-4;(x) = a (t) 0.(x) +az(t). 02(x) +az(t)..(x) (4) Here, aj(t) are unknown parameters, not constants. 0;(x) are shape functions meeting the given boundary conditions: TEX = UN sin (5) (j = 1, 2, 3, ...) u(x, t) = : +4 + + 43 (x, t) = .. : +.. 02 +s. Da '(xt) = a, 01 + a2 0 +a, da " (xt) = a1.0" +22.0%+a, 0 (6) Dj sin MIX 0;=j, COS (7) j22 JTEX 0;' = II = Jei visin (') = - and 50:- o; dx = So for i =] (1 for i =) (8) 8 Weight function w(x) must be 0(x)