initial and boundry condition

consider a 1d heat equation ∝∂^2(T)/∂(x^2)=∂(T)/∂(t)

in the above 1d heat equation T represents temperature on a body in which we applying the equation T=T(x,t) , ∝ is thermal diffusivity

in order to solve the above equation we need some set of conditions and assumptions.

• initial condition are set of conditions that define the behaviour of above equation in the boundry of domian i.e(diffusion,radiation,convection)
• boundry conditions are set of constraints that define the value of function at initial time i.e(inlet,outlet temperature)
• boundry conditions varies with type of pde equation for elliptic we hav elliptic operator whereas for hyperbolic we hav hyperbolic operator
• in hyperbolic pde ,the smoothness of the soluton depends on the smoothness of initial and boundry conditions
• in elliptic pde the solution is always smooth even if those conditions are rough
• in parabolic pde the solution is timedependent i.e(solution are smooth in space but may posses singularities
• type of pde is known by discriminant (b^2-4ac)
• if d=1 parabolic pde
• if d>1 hyperbolic pde
• if d<1 elliptic pde
• the above heat equation is parabolic