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PARTIAL DIFFERENTIAL EQUATIONS
(PDE) FOR ENGINEERING ACTUAL EXAM
WITH ALL POSSIBLE QUESTIONS AND
100% CORRECT SOLUTIONS/A GRADE
What is the order of a differential equation? - ANSWER The order of a DE is the order of the highest derivative that appears in the equation What is the degree of the differential equation? - ANSWER The degree of the DE is the power of the highest derivative involved When is a differential equation linear? - ANSWER A DE is linear if the function F, such that F(x,t,y(x,t)),dy/dx(x,t),dy/dt(x,t),...) is a linear function of y and its derivatives What is cosh(x) and sinh(x) in terms of the exponential function? - ANSWER cosh(x) = (e^x + e^-x)/ sinh(x) = (e^x - e^-x)/ What are the different types/methods of solving first order ODEs? (5)` - ANSWER (1) Direct Integration (2) Seperable Variables (3) Homogeneous 1st-order ODE (4) Linear 1st-order ODE (5) Exact 1st-order ODE
What is the form of a homogeneous 1st-order ODE? How do we solve them? - ANSWER dy/dx = g(y/x) for some function g. We can solve by the change of variables z = (y/x) (SEE NOTES FOR MORE) What is the form of a linear 1st-order ODE? How do we solve them? - ANSWER (dy/dx)
- p(x)y = r(x) This can be solved using an integration factor. Let P(x) be an antiderivative of p(x) and then multiply through by e^P(x), then solve by direct integration What is the form of an exact 1st-order ODE? How do we solve them? What are the necessary conditions? - ANSWER (dy/dx) = (-M(x,y))/N(x,y) or, by rearranging M(x,y) + N(x,y)(dy/dx) = 0 This equation is defined to be exact if there exists a function u(x,y) such that: ∂u/dx = M(x,y) and ∂u/dy = N(x,y) Hence, the solution is (implicitly) given by u(x,y) = constant A necessary and sufficient condition for an equation to be exact is : ∂M/dy = ∂N/dx What is the existence and uniqueness theorem for 1st order derivatives? - ANSWER If f(y,x) and ∂f/∂y are continuous functions for a (2) complex roots; (3) repeated roots? - ANSWER (1) y = Ae^(v_1t) + Be^(v_2t) (2) If v = α ± iβ y = e^(αt)[Acos(βt) + Bsin(βt)] (3) y = Ae^(vt) + Bt.e^(vt) What form do the Euler type of linear, homogeneous equations have? - ANSWER ax^2.d^2y/dx^2 + bx.dy/dx + cy = 0 What is the Ansatz we make for the Euler type equation? - ANSWER y = x^v solve for v What is the auxiliary equation for Euler type ODE? - ANSWER av^2 + (b-a)v + c = 0 What is the general solution of the homogeneous Euler type ODE if the auxiliary equation has: (1) real roots; (2) complex roots; (3) repeated roots? - ANSWER (1) y = Ax^v_1 + Bx^v_ (2) v = α ± β y = x^α[Acos(βln(x)) + Bsin(βln(x))] (3) y = x^v[A + Bln(x)] What is the existence and uniqueness theorem for second order ODEs? - ANSWER Given a function f(y_2,y_1,t), suppose that f, ∂f/∂y_1 and ∂f/∂y_2 are continuous functions for a_1 < y_1 < a_2, b_1 < y_2 < b_2 and t_1 < t < t_2. Then for all initial conditions: y(t_0) = y_0 and y'(t_0) = u_
with a_1 < y_0 < a_2, b_1 < u_0 < b_2 and t_1 < t < t_2, there exists a unique solution of y'' = f(y', y, t) on some interval I containing t_ What is the Wronskian? - ANSWER Given two differentiable functions y_1,y_2, their Wronskian is defined by the determinant: W[y_1(x),y_2(x)] = y_1(x).y'_2(x) - y_2(x)y'_1(x) If the Wronksian of two differentiable functions does not vanish at some point x_0, then what do we know about these functions? - ANSWER The functions are linearly independent Given the homogeneous 2nd order linear ODE with p(x) and q(x) continuous in an interval I, two solutions y_1 and y_2 of this ODE are: (1) linearly independent on I iiff; (2) linearly dependent on I iff? - ANSWER (1) Are linearly independent on I if and only if W(x_0) =/= 0 for some x_0 in I (2) Are linearly dependent on I if and only if W(x_0) = 0 for some x_0 in I Consider any two function sy_1 and y_2 that are solutions of the homogeneous 2nd order ODE with p(x) and q(x) are continuous in I. Then the Wronskian of y_1 and y_2 is either ______ or _________? - ANSWER Then the Wronskian of y_1 and y_2 is either identically zero for all x in I, or never zero for all x in I If the ODE doesn't depend on y explicitly, what substitution will give us a first order ODE for z as a function of x? - ANSWER z = y' If the ODE doesn't depend on x explicitly, what substitution will give us a first order ODE for z as a function of y? - ANSWER z = y'
What do we choose to avoid of v_1 and v_2 in variation of parameters? - ANSWER We choose to avoid second dervatives of v_1 and v_2 sp we set: v'_1.y_1 + v'2.y_2 = 0 What are the steps in the variation of parameters method? (4) - ANSWER (1) Compute the complementary functions y_1 and y (2) Compute v_1 = - ∫[r(x).y_2(x)]/Wy_1,y_2dx v_2 = ∫[r(x).y_1(x)]/Wy_1,y_2dx where W[y_1,y_2] is the Wronskian (3) Compute the particular integral y_p(x) = v_1(x).y_1(x) + v_2(x)y_2(x) (4) Write down the general solution y(x) = c_1.y_1(x) + c_2.y_2(x) + y_p(x) What are the 3 different cases we get with boundary value problems depending on the choice of domain and the other boundary condition? - ANSWER (1) Unique solution (2) No solution (3) Infinitely many solutions How can we write the boundary conditions in a general form? - ANSWER a_0.y(x_0) + a_1.y'(x_0) = α b_0.y(x_1) + b_1.y'(x_1) = β What are the three cases we split the eigenvalue problem into? - ANSWER (1) λ = 0 (2) λ = k^2 > 0
(3) λ = - k^2 < 0 What do we do in eigenvalue problems? - ANSWER The problem to solve involves finding the conditions under which the problem has a solution. We begin by looking for NON-TRIVIAL solutions of the simple BVP with unknown parameter λ What are the eigenvalues? What are the eigenfunctions? - ANSWER The set of λ of which the ODE admits non- trivial solutions are called eigenvalues. The corresponding solutions are called the eigenfunctions. What is the Sturm-Liouville form? - ANSWER y'' + b(x)y' + c(x)y = λd(x)y then, (p(x)y')' + q(x)y - λw(x)y = 0 where p(x) = e^B(x), q(x) = c(x)e^B(x), w(x) = d(x)e^B(x) What is the linear differential operator L in the Sturm-Liouville form? - ANSWER L: y - > L[y] = (1/w(x)){(p.y')' + q.y} What is an inner product with weight function w(x) on two functions f and g? - ANSWER An inner product with weight function w(x) on two functions f and g defined on the interval is: _w = ∫f(x)g(x)w(x)dx A solution y_n of the regular Sturm-Liouville problem is called what? What is the associated vlaue of λ called? - ANSWER A eigenfunction The associated value of λ, λ_n is the corresponding eigenvalue
- q(x) ≤ 0 for x∈[x_0,x_1]
- p(x_0)y'(x_0)y(x_0) - p(x_1)y'(x_1)y(x_1) = 0 What is the error ε_N? - ANSWER ε_N = ∫w(x)[F(x) - ∑a_n.y_n(x)]^2.dx How can we expand a periodic function of period 2π Fourier Series? - ANSWER f(x) = (1/2).a_0 + ∑(a_n.cos(nx) + b_n.sin(nx)) where a_n and b_n are constants called the Fourier constants and are such that the right hand side converges for all x How do we write the Fourier Series for a periodic function of period 2L? - ANSWER f(x) = (1/2).a_0 + ∑(a_n.cos(nπx/L) + b_n.sin(nπx/L)) What are some important indentites of sin? (3) - ANSWER (1) sin(nπ) = 0 (2) sin((n + 0.5)π) = (-1)^n (3) 2sinAsinB = cos(A - B) - cos(A + B) What are some important identities of cos? (4) - ANSWER (1) cos(nπ) = (-1)^n (2) cos((n+0.5)π) = 0 (3) cos(A + B) = cosAcosB - sinAsinB (4) cos(A - B) = cosAcosB + sinAsinB What does δ_mn equal? - ANSWER δ_mn = {0 if m=n, 1 if m=/= n} What are the integral results if the function has period 2L? What if m= 0? - ANSWER (1) ∫sin(mπx/L)sin(nπx/L)dx = L.δ_mn
(2) ∫cos(mπx/L)cos(nπx/L)dx = L.δ_mn (3) ∫sin(mπx/L)cos(nπx/L)dx = 0 IF m=0 the cos relation doesn't hold, in this case; ∫cos(nπx/L)dx = 2L.δ_mn What are the integral results if the function has period 2π? - ANSWER (1) ∫sin(mx)sin(nx)dx = π.δ_mn (2) ∫cos(mx)cos(nx)dx = π.δ_mn (3) ∫sin(mx)cos(nx)dx = 0 What are the Euler Formulae if the period is: (1) 2L; (2) 2π? - ANSWER (1) a_m = (1/L)∫f(x).cos(mπx/L)dx b_m = (1/L)∫f(x).sin(mπx/L)dx (2) a_m = (1/π)∫f(x).cos(mx)dx b_m = (1/π)∫f(x).sin(mx)dx When is a function f an: (i) even function; (ii) odd function? - ANSWER (i) f(-x) = f(x) i.e symmetric about y-axis (ii) f(-x) = - f(x) What is the integral of f(x) over a symmetric domain equal to if f is: (i) even;
What is an even periodic extension? - ANSWER Suppose that f(x) is given only on 0≤x≤L. Then we can extend the function onto - L ≤ x ≤ L by making f(-x) = f(x). We then extend this to all values of x by making the function have period 2L What are the Dirichlet's conditons for a Fourier series? - ANSWER (1) f(x) is bounded (2) The function is 2L-periodic for some L (3) In any period, f(x) has a finite number of maxima, minima and discontinuities What is the Fourier Convergence Theorem? - ANSWER Let f(x) be a bounded, 2L-period function with a finite number of maxima, minima and discontinuities in any period. Then the sum (1/2).a_0 + ∑(a_n.cos(nπx/L) + b_n.sin(nπx/L)) converges for all finite x, and when a_n,b_n are given by the Euler formulas the sum converges to f(x) if, at x, f is continuous. At points where f is not continuous the sum converges to (1/2)[f(x-) + f(x+)] where f(x-) and f(x+) are the values of the function as we approach from the left and right respectively. What is the other ways of writing: (1) 2sin(mx)sin(nx); (2) 2cos(mx)cos(nx); (3) 2sin(mx)cos(nx)? - ANSWER (1) cos(m-n)x - cos(m+n)x (2) cos(m-n)x + cos(m+n)x (3) sin(m+n)x + sin(m-n)x What is the derivative of a Fourier series f(x)? When does this hold?(2) - ANSWER f'(x) = ∑[(-n.π.a_n/L)sin(nπx/L) + (n.π.b_n/L)cos(nπx/L)] This only holds when:
(1) f(x) is continuous everywhere (2) f'(x) satisfies the Dirichlet's conditions Can we always integrate the Fourier seires? Does it give a Fourier series? - ANSWER The Fourier series may ALWAYS be integrated term by term. If it contains an 'x' term it is not a Fourier series, but we only need to replace x with the Fourier series for x What kind of PDE do we get if: (1) b^2 > 4ac; (2) b^2 = 4ac; (3) b^2 < 4ac? Give an example for each one. - ANSWER (1) Hyperbolic. eg Wave equation (2) Parabolic. e.g diffusion equation (3) Elliptic. e.g Laplace's equation What is the general form of a second order linear PDE? - ANSWER a(x,y)∂^2u/∂x^2 + b(x,y)∂^2u/∂xy + c(x,y)∂^2u/∂y^2 + d(x,y)∂u/∂x + e(x,y)∂u/∂y + f(x,y).u = 0 What is the wave equation? - ANSWER ∂^2y/∂t^2 = c^2.∂^2y/∂x^ where c is a constant What is the change of variables we make in D'Alembert's solution? When we substitute it into the wave equation what does it become? - ANSWER ξ = x + c.t η = x - c.t
What are the normal mode solutions? What do they mean? - ANSWER y_n(x,t) = [C_n.cos(nπct/L) + D_n.sin(nπct/L)].sin(nπx/L) They're the set of solutions which only satisfy the wave equation and boundary conditions but not initial conditions. What is the general solution of the wave equation using the superimposed normal mode solutions? - ANSWER Y_n(x,t) = ∑[C_n.cos(nπct/L) + D_n.sin(nπct/L)].sin(nπx/L) How do we find the coefficients C_n and D_n in the general solution of the wave equation? - ANSWER Evaluating our general solution at t=0 gives: p(x) = ∑C_n.sin(nπx/L) (Fourier sine series for C_n) Evaluating the time derivative of the general solution at t=0 gives: q(x) = ∑D_n.(nπc/L)sin(nπx/L) (Fourier sine series for D_n) Using Euler formulae we get: C_n = (2/L)∫p(x).sin(nπx/L)dx D_n = (2/nπc)∫q(x).sin(nπx/L)dx (Where the integration interval is 0 to L) What seperation of variable in 6 steps, where y(x,t) = X(x).T(t)? - ANSWER (1) Determine equations for X,T (2) Use boundary conditions of y in order to obtain boundary conditions of X (3) Solve eigenvalue problem for X to get λ_n and X_n (4) Inset λ_n into T and solve to get T_n (5) The normal modes are y_n = X_n.T_n and the general solution obtained by superposition y(x,t) = ∑X_n(x).T_n(t) (6) Use the initial conditions y(x,0), ∂y(x,0)/∂t to determine all undertimed coefficients. This step involves Fourier series
In Cartesian coordinates, what is the heat equation in: (1) One dimension; (2) Two dimensions; (3) Three dimensions? - ANSWER (1) ∂u/∂t = κ^2.(∂^2u/∂x^2) (2) ∂u/∂t = κ^2.(∂^2u/∂x^2 + ∂^2u/∂y^2) (3) ∂u/∂t = κ^2.(∂^2u/∂x^2 + ∂^2u/∂y^2 + ∂^2u/∂z^2) If we have the heat equation and Neumann or Dirichlet boundary conditons, how can we solve it? - ANSWER Using the normal Ansatz in seperation of variables, y = XT and do the same method as with wave equation What is the two dimensional heat equation in polar coordinates? How can we solve it? - ANSWER ∂u/∂t = κ^2.(∂^2u/∂r^2 + (2/r).∂u/∂r) We can solve it using seperation of variables What is the inhomogeneous form of the heat equation? What is the source term? - ANSWER ∂y/∂t = κ^2.(∂^2y/∂x^2) + F(x,t) where F(x,t) is the source term How do we solve the inhomogeneous form of the heat equation, with source term F(x,t), if we have Dirichlet boundary conditions y(0,t) = 0 = y(1,t)? - ANSWER (1) We make the assumption that spacial dependence of the solution and source term can be described in terms of the solution of the corresponding source free problem (2) So we get that y(x,t) = ΣT_n(t).sin(nπx) and F(x,t) = ΣF_n(t).sin(nπx) ,where F_n(t) is the coefficient in the sine Fourier series of F(x,t) (3) Now sub our expressions into the original heat equation to get: Σ[T'_n(t) + (κn π)^2.T_n(t) - F_n(t)]sin(nπx) = 0, hence T'_n(t) + (κn
When we have the heat equation with inhomogeneous BC's what are the Dirichlet BC of v(x,t)? - ANSWER v(0,t) = 0 = v(1,t) How do we solve the standard heat equation with inhomogeneous boundary condtions y(0,t) = f(t) and y(1,t) = g(t)? - ANSWER (1) We make the change of variables y - > v using y(x,t) = v(x,t) + y_p(x,t) and sub into the heat equation, where y_p(x,t) = x.g(t) + (1-x).f(t) (2) So we get, ∂y_p/∂t + ∂v/∂t = κ^2.(∂^2v/∂t^2) with Dirichlet BC: v(0,t) = 0 = v(1,t) (3) This gives us the inhomogeneous heat equation: ∂v/∂t = κ^2.(∂^2v/∂t^2) + F(x,t) where F(x,t) = - ∂y_p/∂t = - x.g'(t) - (1-x).f'(t) (4) Then we solve the inhomogeneous problem as normal, solving for v and then using this to hence find y What is the heat equation in spherical polars? - ANSWER ∂y/∂t = κ^2.(1/r^2).∂/∂r(r^2.∂y/∂r) How can the heat equation be generalised? - ANSWER w(x)∂y/∂t = ∂/∂x(p(x)∂y/∂x) + q(x)y Our standard seperation of variables ansatz y = XT in the generalised heat equations gives what? - ANSWER T* - λT = 0 (p(x)X')' + q(x)X = λw(x)X Where ' is differentiation w.r.t x and * is w.r.t t
This gives the eigenvalue problem: (p(x)X')' + q(x)X = λw(x)X What does it mean if the differential operator L is self-ajoint w.r.t the weight function w ? What do we need for the problem to be self-adjoint? - ANSWER (1) We know that L is is self-adjoint w.r.t to w if: ∫y_A.L(y_B).w(x)dx = ∫y_B.L(y_A).w(x)dx (between b and a), where y_A and y_B are solutions satisfying boundary conditions (2) For the problem to be self-adjoint we need: [p(x).y_A.(y_B)' - p(x).y_B.(y_A)'] = 0 (between a and b) What are the essential properties we need for a Sturm-Liouville problem? (PDE context) (3) - ANSWER (1) The real eigenvalues λ_n are distinct (2) The associated eigenfunctions y_n are orthogonal in the sense: ∫w(x).y_n(x).y_m(x)dx = 0, m=/=n (3) If f(x) is bounded and piecewise continuous on [a,b], then we can write: f(x) = ΣC_n.y_n(x) and the eigenfunctions y_n form a basis for a suitable Hilbert space What is Laplace's equation in Cartesian coordinates and in: (i) R ; (ii) R^2 ; (iii) R^3? - ANSWER (i) d^2φ/dx^ (ii) (∂^2/∂x^2 + ∂^2/∂y^2)φ (iii) (∂^2/∂x^2 + ∂^2/∂y^2 + ∂^2/∂z^2)φ
