ES305, COMPUTING METHODS IN ENGINEERING, FALL SEMESTER
1) Solve the
following differential equation using h=0.1 for two steps
_{}
a) by Euler’s
method,
b) by Modified Euler’s
(implicit) method
c) compare your
results found in parts “a” and “b” and comment on the differences if any.
2) The governing differential equation of a SDOF (single
degree of freedom) system given below with the corresponding initial conditions
as
_{}; x(0)=0, _{}where _{}.
Find the displacements at t=
.01 and 0.02 seconds (i.e. h=_{}=0.01)
by using
i)
Euler’s method for systems
ii)
Runge Kutta’s method of order 2 for systems.
3) Consider,
Governing Equation (GE) : _{}
with the boundary Conditions (BC's): y
(0) = 0, y (1) = 0
i)
Solve the second order ordinary differential equation
given above by Shooting method by subdividing the solution region 0 £ x £ 1
into 4 equal parts.
ii)
Compare the results found in (i) with the results of
the exact solution of this BVP given as _{}
4) Given the following
boundary value problem
_{}_{}
a) Show that the
following matrix equation is the reduced form of the above boundary value problem by finite differences of order h^{2} with h =0.25
_{}
b) Solve the above matrix equation in part ‘a’ by LU
decomposition.
5)

Figure shown on the left is the simple beam with the length of L, Modulus of Elasticity of E and Moment of Inertia of I. It is subjected to the uniformly distributed load q. Governing equation for this beam is:
_{} and B.C. at x =
0 v = 0, _{}
at x = L v = 0, _{}
Where v is the
lateral displacement and x is the horizontal axis. By using the formulas
_{} and _{} ,
obtain the linear
system of equations for six divisions in terms of q, L, E, I. Solve the system
by G.E.M. and compare the maximum displacement with the
_{}
(Find the actual
error). At first, what can you do to decrease an error?