Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8

Second Semester 1442 Maximim Time: Three Hours

Name:

Student number:

Field of specialization:

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8

Note : Please answer 4 questions of 8.

Q1) Solve the initial value problem

y⁰ =−2ty², y(0) = 1

with h= 0.2 on the interval [0,1].Use the fourth order Runge Kutta method

yi+1 = yi+ 1

6(K1+ 2K2+ 2K3+K4) K₁ = hf(t_i, y_i)

K₂ = hf(t_i+h

2, y_i+K₁) K₃ = hf(t_i+h

2, y_i+K₂) K₄ = hf(t_i+h, y_i+K₃)

to approximatey(0.2),Compare your answer with the exact solution y(t) = _t2¹+1

b)Use Gauss elimination method to solve the linear system

2x₁ + 4x₂−x₃ = −5 x₁ +x₂−3x₃ = −9 4x₁+x₂ + 2x₃ = 9

Q2) Consider the nonlinear system dx

dt =−x+ 2xy−x², dy

dt =y(1−y).

1. Find all the equilibrium points.

2. Calculate the Jacobian matrix.

3. Calculate the eigenvalues of the Jacobian matrix at each equilibrium.

4. Based on the result, classify the equilibrium point into one of the following: Stable node, unstable node, saddle point, stable focus, unstable focus, or center.

Q3) (a) Consider the boundary value problem:

x⁰⁰(t) = g(t, x(t)), t ∈[0,1], x(0) = 0, x(1) = 1.

Find a single integral representing the solution of the equation.

(b) Consider the diffusion equation

u_t =νu_xx, t >0, 0< x≤π,

u(0, t) =u(π, t) = 0, t >0 u(x,0) =f(x), 0< x≤π.

By using an appropriate method, find the solution.

Q4) (a) Show that the function f(z) = |z|² is differentiable only at z = 0, and compute f⁰(0).

(b) Find the Laurent series about z = 0 for f(z) = _z2−3z+2⁻¹ in 1<|z|<2.

(c) EvaluateR

C e^2z

(z+2)²dz, whereC is the circle |z|= 3

Q5)

a) Prove the H ={i,−i,1,−1} is a finitely generated abelian group, where i=√

−1.

b) Consider the additive finite abelian group Z12 and the cyclic subgroup H = {0,4,8}

of Z12. Using Lagrange’s Theorem, find the index of H inZ12 ( [Z12 :H] = ).

c) Let V = R³ be a vector space over the field of real numbers R and let x = (1,0,0), y= (1,1,0) andz = (1,1,1). Show that V =R_x+R_y+R_z.

Q6) Find a topological space and a compact subset whose closure is not compact. Note that a space is compact if every open cover of it has a finite subcover.

Q7) (a) Find a sequence which converges to zero but is not in any`<sup>p</sup> space, 1≤p <∞, and prove that the space `^p is separable.

(b) Prove that if the dual space X⁰ of a normed space X is separable, then X itself is separable.

Q8) (a) Let {f_n} be an increasing sequence of nonnegative measurable functions defined on a measurable set E, and let f = lim

n→∞f_n a.e. Prove that Z

E

f = lim

n→∞

Z

E

fn.

(b) LetM be aσ-algebra of sets of real numbers. Define m:M→ [0,∞] by m(E) =

∞ if E 6=∅ 0 if E =∅.

Is m a countably additive measure? Justify.

(c) Letm^∗ be the Lebesgue outer measure. Show that if A is countable, then m^∗(A) = 0.