Question

• Set up a study about Partial Differential Equation Equations (PDEs)
• Discuss types and methods of solving the Differential Equation Equations (PDEs)
• Elliptic Equations
• Parabola equation
• Hyperbola equation
• Critically analyze the three PDEs indicating the possible boundary condition and make a comparison between them
• Indicate how to drive the three equations : Explicit, implicit, and crank Nicolson

Answer 1

Solutions partial differential equations (Poe's) Poe's are the most cornmon method by which we model physical problems in engineering. finite elements methods are one of many ways of . solving PDE'S. definition: O A PDE is a relationship between an unknown Junction of several variables and its partial derivatives. an unknown Let uld, s, zit) be function. u= u(x,x2, Bat) a ie the dependent variable partial derivatives are denoted by su = 3²4 Ja, ja ", a Edu , usa g²4 ja,? 2«, u j²u 12 18,18,

Examples: l. Laplace equation : Ja + sio = 0 sse? . 582 . J 8² 2 poisson's equation : 0p=el where e= source density. tion - 3. Head slow_cquation of the Heat 52 where h²= diffusition. linear is oft is of first degree & non-linear PRES : eos PoE A linear is is one one that that of its field variables and in all partial dexiva tering, fos &q : hoe + o iſ lincon su Atsu Iso is non-linear. Sa t \oda

Homogeneous POE'S :- If $(&, 82, 83,t) =0 the Poe is called homogeneous. i. Ju + suot su + Ju zo is homogeneous. Stt Jo, jo, saz su & su t Jt. Ja, u + sua att jag. Jag iã non- homoge -neous. Elliptic, hyperbolic and para bolic PoES : there are 3 types of second order poɛ in mechanics. these are classified as ellistic, hyperbolic and parabolic, the equations of elasticity are ellistic Poe's hyperbolic Poe's describe ware propagation - the heat conduction equation is parabolic PPE. Suppose we have second order poe of the form a(, 42) septie + b (*, 2) + 6(,y) on the + a (4,) enero + € (4,89) t $(,,,) v = g (4, ) then PDE is called elliptic it beruac co

the Poe is called hyperbolic if (6²_nac sol - ya the PRE is called parabolic it 16²-4ac :o the crank - Nicolson me thod: It is based on tsafe.zoidal sule giving and order convergence in time. for example su o ff (u,a,t, su, Letting ulinx, nat) and Rinaç uin uin. evaluated for in and ia a tinite difference method this solving method heat for equation. Implicit Vs Explicit methods:- Implicit methods : formula at each fast. a) explscit there ig¢ no which only equations a set of simultaneous solved over the must be whole grida are stable for all step - Implicit methods sizes.

Explisit methods - possible to solve directly for all unknown values in the finite difference Scheme. - - . Stable only for certain time step sizes stability can be checked using Jourier or von neumann analysis. time step size. governed by cocusent current condition for cans q.