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- Question : EX1 - Classify each of the partial differential equations below as either hyperbolic, parabolic, or elliptic, determine the characteristics, and transform the equations to canonical form: (a) 4uxx + 5uxy + uyy + ux + uy = 2, (b) 2uxx ? 3uxy + uyy = y, (c) yuxx + (x + y)uxy + xuyy = 0, (d) uxx + yuyy = 0, (e) yuxx ? 2uxy + exuyy + x2ux ? u = 0, (f) uxx + xuyy = 0, (g) x2uxx + 4yuxy + uyy + 2ux = 0, (h) 3yuxx ? xuyy = 0, (i) uxx + 2xuxy + a2uyy + u = 5, (j) y2uxx + x2uyy = 0.
- Question : EX2 - Classify each of the partial differential equations below as either hyperbolic, parabolic, or elliptic, determine the characteristics, and transform the equations to canonical form: (a) 4uxx + 5uxy + uyy + ux + uy = 2, (b) 2uxx ? 3uxy + uyy = y, (c) yuxx + (x + y)uxy + xuyy = 0, (d) uxx + yuyy = 0, (e) yuxx ? 2uxy + exuyy + x2ux ? u = 0, (f) uxx + xuyy = 0, (g) x2uxx + 4yuxy + uyy + 2ux = 0, (h) 3yuxx ? xuyy = 0, (i) uxx + 2xuxy + a2uyy + u = 5, (j) y2uxx + x2uyy = 0.
- Question : EX3 - Determine the nature of the following equations and reduce them to canonical form: (a) x2uxx + 4xyuxy + y2uyy = 0, (b) uxx ? xuyy = 0, (c) uxx ? 2uxy + 3uyy + 24uy + 5u = 0, (d) uxx + sech4xuyy = 0, (e) uxx + 6yuxy + 9y2uyy + 4u = 0, (f) uxx ? sech4xuyy = 0, (g) uxx + 2cosecyuxy + cosec2yuyy = 0, (h) uxx ? 5uxy + 5uyy = 0.
- Question : EX4 - (a) Show that the nonlinear equation u2uxx + 2uxuyuxy ? u2uyy = 0 is hyperbolic for every solution u(x, y). (b) Show that the nonlinear equation for the velocity potential u(x, y) ?1 ? u2x?uxx ? 2uxuyuxy + ?1 ? u2y?uyy = 0 in certain kinds of compressible ?uid ?ow is (i) elliptic, (ii) parabolic, or (iii) hyperbolic for those solutions such that |?u| < 1, |?u| = 1, or |?u| > 1.
- Question : EX5 - SOLVE THAT THE NONLINEAR EQUATION FOR THE VELOCITY POTENTIAL
- Question : EX6 - Use the separation of variables to solve the Laplace equation uxx + uyy = 0, 0 ? x ? a, 0 ? y ? b, with u(0, y) = 0 = u(a, y) for 0 ? y ? b, and u(x, 0) = f(x) for 0 < x
- Question : EX7 - SHOW THE EIGNVALUES OF THE EIGNVALUES PROBLEM
- Question : EX8 - Solve the problem in Exercise 4 with the boundary conditions
- Question : EX9 - Solve Example 1.6.2 with the initial data
- Question : EX10 - Solve Example 1.6.1 with the initial data (i) f(x) = ? hax if 0 ? x ? a, h(? ? x)/(? ? a) if a ? x ? ?, and g(x) = 0. (ii) f(x) = 0 and g(x) = ? ua0x if 0 ? x ? a, u0(? ? x)/(? ? a) if a ? x ? ?
- Question : EX11 - Solve Example 1.6.1 with the initial data (i) f(x) = ? hax if 0 ? x ? a, h(? ? x)/(? ? a) if a ? x ? ?, and g(x) = 0. (ii) f(x) = 0 and g(x) = ? ua0x if 0 ? x ? a, u0(? ? x)/(? ? a) if a ? x ? ?
- Question : EX12 - Find the solution of the dissipative wave equation
- Question : EX13 - Solve the Cauchy problem for the linear Klein
- Question : EX14 - Solve the telegraph equation
- Question : EX15 - The transverse vibration of an in?nite elastic beam of mass m per unit length and bending stiffness EI is governed by utt + a2uxxxx = 0, a2 = EI m , ?? < x < ?, t > 0. Solve this equation subject to the boundary and initial data u(0, t) = 0 for all t > 0, u(x, 0) = ?(x), and ut(x, 0) = ???(x) for 0 < x < ?. Show that the Fourier transform solution is U(k, t) = ?(k) cos ?atk2? ? ?1a??(k) sin?atk2?. Find the integral solution for u(x, t).
- Question : EX16 - Solve the Lamb (1904) problem in geophysics that satis?es the Helmholtz equation in an in?nite elastic half-space uxx + uzz + ?2 c2 2 u = 0, ?? < x < ?, z > 0, where ? is the frequency and c2 is the shear wave speed. At the surface of the half-space (z = 0), the boundary condition relating the surface stress to the impulsive point load distribution is given by
- Question : EX17 - Find the solution of the Cauchy
- Question : EX18 - Obtain the solutions for the velocity potential ?(x, z, t) and the free surface elevation ?(x, t) involved in the two-dimensional surface waves in water of ?nite (or in?nite) depth h. The governing equation, boundary, and free surface conditions and initial conditions (see Debnath 1994, p. 92) are
- Question : EX19 - Solve the steady-state surface wave problem (Debnath 1994, p. 47) on a run- ning stream of in?nite depth due to an external steady pressure applied to the free surface. The governing equation and the free surface conditions are ?xx + ?zz = 0, ?? < x < ?,?? < z < 0, t > 0, ?x + U?x + g? = ?P? ?(x) exp(?t), ?t + U?x = ?z ? onz = 0(? > 0), ?z ? 0 as z ? ??. where U is the stream velocity, ?(x, z, t) is the velocity potential, and ?(x, t) is the free surface elevation.
- Question : EX20 - Apply the Fourier transform to solve the initial-value problem for the dissipa- tive wave equation utt = c2uxx + ?uxxt, ?? < x < ?, t > 0, u(x, 0) = f(x), ut(x, 0) = ?f??(x) for ? ? < x < ?, where ? is a positive constant.
- Question : EX21 - Use the Fourier transform to solve the boundary-value problem uxx + uyy = ?xexp??x2?, ?? < x < ?, 0 < y < ?,
- Question : EX22 - Solve the initial-value problem (Debnath 1994, p. 115) for the two-dimensional surface waves at the free surface of a running stream of velocity U. The prob- lem satis?es the following equation, boundary, and initial conditions: ?xx + ?zz = 0, ?? < x < ?,?h ? z ? 0, t > 0, ?x + U?x + g? = ?P? ?(x) exp(i?t), ?t + U?x ? ?z = 0 ? onz =0,t>0, ?(x, z, 0) = ?(x, 0) = 0, for all x and z.
- Question : EX23 - Apply the Fourier transform to solve the equation uxxxx + uyy = 0, ?? < x < ?, y ? 0, satisfying the conditions u(x, 0) = f(x), uy(x, 0) = 0 for ? ? < x < ?,
- Question : EX24 - The transverse vibration of an in?nite elastic beam of mass m per unit length and bending stiffness EI is governed by
- Question : EX25 - Solve the diffusion problem with a source function q(x, t) ut = ?uxx + q(x,t), ?? < x < ?, t > 0, u(x,0) = 0 for ? ? < x < ?. Show that the solution is u(x, t) = 1 ?4???t 0 (t??)?21d?? ? ?? q(k,?)exp?? (x?k)2 4?(t ? ?) ?dk
- Question : EX26 - Apply the triple Fourier transform to solve the initial-value problem ut = ?(uxx + uyy + uzz), ?? < x, y, z < ?, t > 0, u(x, 0) = f(x) for all x, y, z, where x = (x, y, z).
- Question : EX27 - Use the double Fourier transform to solve the telegraph equation utt + aut + bu = c2uxx, ?? < x,t < ?, u(0, t) = f(t), ux(0, t) = g(t), for ? ? < t < ?, where a, b, c are constants and f(t) and g(t) are arbitrary functions of t.
- Question : EX28 - Use the Fourier transform to solve the Rossby wave problem in an inviscid ?- plane ocean bounded by walls at y = 0 and y = 1, where y and x represent vertical and horizontal directions. The ?uid is initially at rest and then, at t = 0+, an arbitrary disturbance localized to the vicinity of x = 0 is applied to generate Rossby waves. This problem satis?es the Rossby wave equation ? ?t???2 ? ?2??? + ??x = 0, ?? < x < ?,0 ? y ? 1, t > 0, with the boundary and initial conditions ?x(x, y) = 0 for 0 < x < ?, y = 0 and y = 1, ?(x, y, t) = ?0(x, y) at t = 0 for all x and y.
- Question : EX29 - The equations for the current I(x, t) and potential V (x, t) at a point x and time t of a transmission line containing resistance R, inductance L, capacitance C, and leakage inductance G are LIt + RI = ?Vx, and CVt + GV = ?Ix. Show that both I and V satisfy the telegraph equation 1 c2 utt ? uxx + aut + bu = 0, where c2 = (LC)?1, a = LG + RC, and b = RG. Solve the telegraph equation for the following cases with R = 0 and G = 0: (a) V (x, t) = V0H(t) at x = 0, t > 0, V (x, t) ? 0 as x ? ?, t > 0, where V0 is constant.
- Question : EX30 - Solve the telegraph equation in Exercise 29 with V (x, 0) = 0 for (a) the Kelvin ideal cable line (L = 0 = G) with the boundary data V (0, t) = V0 = const., V (x, t) ? 0 as x ? ? for t > 0. (b) the noninductive leaky cable (L = 0) with the boundary conditions V (0, t) = H(t) and V (x, t) ? 0 as x ? ? for t > 0.
- Question : EX31 - Solve the telegraph equation in Exercise 29 with V (x, 0) = 0 = Vt(x, 0) for the Heaviside distortionless cable (RL = CG = const. = k) with the boundary data V (0, t) = V0f(t) and V (x, t) ? 0 as x ? ? for t > 0, where V0 is constant and f(t) is an arbitrary function of t. Explain the physical signi?cance of the solution.
- Question : EX32 - Solve the inhomogeneous partial differential equation
- Question : EX33 - Find the solution of the inhomogeneous equation 1 c2 utt ? uxx = k sin??ax?, 0 < x < a, t > 0, u(x, 0) = 0 = ut(x, 0) for 0 < x < a, u(0, t) = 0 = u(a, t) for t > 0.
- Question : EX34 - Solve the Stokes problem which is concerned with the unsteady boundary layer ?ows induced in a semi-in?nite viscous ?uid bounded by an in?nite horizontal disk at z = 0 due to nontorsional oscillations of the disk in its own plane with a given frequency ?. The equation of motion and the boundary and initial conditions are ut = ?uzz, z > 0, t > 0, u(z, t) = Uei?t on z = 0, t > 0, u(z, t) ? 0 as z ? ? for t > 0, u(z, 0) = 0 for t ? 0 and z > 0, where u(z, t) is the velocity of the ?uid of kinematic viscosity ? and U is con- stant. Solve the Rayleigh problem (? = 0). Explain the physical signi?cance of both the Stokes and Rayleigh solutions.
- Question : EX35 - Solve the Blasius problem of an unsteady boundary layer ?ow in a semi-in?nite body of viscous ?uid enclosed by an in?nite horizontal disk at z = 0. The governing equation and the boundary and initial conditions are ?u ?t = ? ?2u ?z2 , z > 0, t > 0, u(z, t) = Ut on z = 0, t > 0, u(z, t) ? 0 as z ? ?, t > 0, u(z, t) = 0 at t ? 0, z > 0.
- Question : EX36 - Obtain the solution of the Stokes
- Question : EX37 - Show that, when ? = 0 in Exercise 36, the steady-?ow ?eld is given by q(z,t)? (a+b)exp???2?i??1/2z?. Hence, determine the thickness of the Ekman layer.
- Question : EX38 - Solve the telegraph equation utt ? c2uxx + 2aut = 0, ?? < x < ?, t > 0, u(x, 0) = 0, ut(x, 0) = g(x), ?? < x < ?.
- Question : EX39 - Use the Laplace transform to solve the initial boundary-value problem ut = c2uxx, 0 < x < a, t > 0, u(x, 0) = x + sin?3?ax? for 0 < x < a, u(0, t) = 0 = u(a, t) for t > 0.
- Question : EX40 - Solve the diffusion equation ut = ?uxx, ?a < x < a, t > 0, u(x, 0) = 1 for ? a < x < a, u(?a, t) = 0 = u(a, t) for t > 0.
- Question : EX41 - Use the joint Laplace and Fourier transform to solve the initial-value problem for transient water waves which satis?es (see Debnath 1994, p. 92) ? 2 ? = ?xx + ?zz = 0, ?? < x < ?,?? < z < 0, t > 0, ?z = ?t, ?t + g? = ?P? p(x)ei?t ? onz =0, t>0, ?(x, z, 0) = 0 = ?(x, 0) for all x and z, where P and ? are constants.
- Question : EX42 - Show that the solution of the boundary-value problem urr + 1 r ur + uzz = 0, 0 < r < ?, 0 < z < ?, u(r, z) = 1 ?a2 +r2 on z = 0, 0 < r < ?,
- Question : EX43 - (a) The axisymmetric initial-value problem is governed by ut = ??urr + 1rur? + ?(t)f(r), 0 < r < ?, t > 0, u(r, 0) = 0 for 0 < r < ?. Show that the formal solution of this problem is u(r,t)=?? 0 kJ0(kr)f
- Question : EX44 - If f(r) = A(a2 + r2)?12 , where A is a constant, show that the solution of the biharmonic equation described in Example 1.10.7 is u(r,z) = A{r2 +(z + a)(2z + a)} [r2 + (z + a)2]3/2 .
- Question : EX45 - Solve the axisymmetric biharmonic equation for the free vibration of an elastic disk b2??2 ?r2 + 1 r ? ?r ? 2 u + utt = 0, 0 < r < ?, t > 0, u(r, 0) = f(r), ut(r, 0) = 0 for 0 < r < ?, whereb2 = D 2?h is the ratio of the ?exural rigidity of the disk and its mass 2h? per unit area.
- Question : EX46 - Show that the zero-order Hankel transform solution of the axisymmetric prob- lem urr + 1 r ur + uzz = 0, 0 < r < ?, ?? < z < ?, lim r?0?r2u? = 0, rli?m0(2?r)ur = ?f(z), ?? < z < ?, is u
- Question : EX47 - Solve the nonhomogeneous diffusion problem ut = ??urr + 1rur? + Q(r,t), 0 < r < ?, t > 0, u(r, 0) = f(r), 0 < r < ?, where ? is a constant.
- Question : EX48 - Solve the problem of the electri?ed unit disk in the (x, t)-plane with center at the origin. The electric potential u(r, z) is axisymmetric and satis?es the boundary-value problem urr + 1 r ur + uzz = 0, 0 < r < ?, 0 < z < ?, u(r, 0) = u0, 0 ? r ? a, ?u ?z = 0, on z = 0 for a < r < ?, u(r, z) ? 0 as z ? ? for all r, where u0 is constant. Show that the solution is u(r, z) = 2u0 ? ?? 0 J0(kr) sin ak k e?kz dk
- Question : EX49 - Solve the axisymmetric surface wave problem in deep water due to an oscilla- tory surface pressure. The governing equations are ? 2 ? = ?rr + 1 r ?r + ?zz = 0, 0 ? r < ?, ?? < z ? 0, ?t + g? = ?P? p(r) exp(i?t), ?z ? ?t = 0 ? onz =0,t>0, ?(r, z, 0) = 0 = ?(r, 0), for 0 ? r < ?, and ?? < z ? 0.
- Question : EX50 - Solve the Neumann problem for the Laplace equation urr + 1 r ur + uzz = 0, 0 < r < ?, 0 < z ? ?, uz(r, 0) = ? 1 ?a2 H(a ? r), 0 < r < ?, u(r, z) ? 0 as z ? ? for 0 < r < ?. Show that lim a?0 u(r, z) = 1 2??r2+z2??21.
- Question : EX51 - Solve the Cauchy problem for the wave equation in a dissipating medium utt + 2?ut = c2?urr + 1rur?, 0 < r < ?, t > 0, u(r, 0) = f(r), ut(r, 0) = g(r) for 0 < r < ?,
- Question : EX52 - Use the joint Laplace and Hankel transform to solve the initial boundary-value problem c2?urr + 1rur + uzz? = utt, 0 < r < ?, 0 < z < ?, t > 0, uz(r, 0, t) = H(a ? r)H(t), 0 < r < ?, t > 0, u(r, z, t) ? 0 as r ? ? and u(r, z, t) ? 0 as z ? ?, u(r, z, t) = 0 = ut(r, z, 0), and show that ut(r, z, t) = ?acH?t ? zc ? ?0? J1(ak)J0?ck?t2 ? zc22 ?J0(kr) dk.
- Question : EX53 - Find the steady temperature u(r, z) in a beam 0 ? r < ?, 0 ? z ? a, when the face z = 0 is kept at temperature u(r, 0) = 0 and the face z = a is insulated except that heat is supplied through a circular hole such that uz(r, a) = H(b ? r). The temperature u(r, z) satis?es the axisymmetric equation urr + 1 r ur + uzz = 0, 0 ? r < ?, 0 ? z ? a.
- Question : EX54 - Find the integral solution of the initial boundary-value problem urr + 1 r ur + uzz = ut, 0 ? r < ?, 0 ? z < ?, t > 0, u(r, z, 0) = 0, for all r and z, ??u ?r ? r=0 = 0, for 0 ? z < ?, t > 0, ??u ?z ? z=0 = ? ?H(a?r) a2 ? r2 , for 0 < r < ?, 0 < t < ?, u(r, z, t) ? 0 as r ? ? or z ? ?.
- Question : EX55 - Solve the Cauchy
- Question : EX56 - Use the joint Hankel and Laplace transform method to solve the initial bound- ary-value problem urr + 1 r ur + utt ? 2?ut = a ?(r) r ?(t), 0 < r < ?, t > 0, u(r, t) ? 0 as r ? ?, u(0, t) is ?nite for t > 0, u(r, 0) = 0 = ut(r, 0) for 0 < r < ?.
- Question : EX57 - Surface waves are generated in an inviscid liquid of in?nite depth due to an explosion (Sen 1963) above it which generates the pressure ?eld p(r, t). The velocity potential ?(r, z, t) satis?es the Laplace equation urr + 1 r ur + uzz = 0, 0 < r < ?, t > 0, and the free surface condition utt + guz = 1 ? ??p ?t ??H(r) ? H?r,r0(t)?? on z = 0, where ? is the constant density of the liquid, r0(t) is the extent of the blast, and the liquid is initially at rest. Solve this problem.
- Question : EX58 - Use the joint Laplace and Fourier transform to show that the solution of the inhomogeneous diffusion problem ut ? ?uxx = q(x, t), x ? R, t > 0, u(x, 0) = f(x) for all x ? R, can be expressed in terms of Green
- Question : EX59 - Find Green
- Question : EX60 - Solve the initial boundary-value problem ut ? ?uxx = q(t)?(x ? V t), x ? R, t > 0, u(x, 0) = 0 and u(x, t) ? 0 as |x| ? ?, where q(t) = 0 for t < 0 and V is constant.
- Question : EX61 - Find the Green function satisfying the equation Gxx + Gyy = ?(x ? ?)?(y ? ?), 0 < x, ? < a, 0 < y, ? < b G(x, y) = 0 on x = 0, and x = a; G(x, y) = 0 on y = 0 and y = b.
- Question : EX62 - Show that the solution of the two-dimensional diffusion equation ut ? ?(uxx + uyy) = q(x, y, t); ?? < x, y < ?, t > 0, with u(x, y; 0) = 0 is u(x,y,t)=? t 0 d???? ?? ? exp??(x ? ?)2 + (y ? ?)2 4?(t ? ?) ?
- Question : EX63 - Find the Green function for the one-dimensional Klein
- Question : EX64 - Use the Fourier series method to solve the equation for a diffusion model ut = ?uxx, ?? < x < ?, t > 0, with the periodic boundary conditions u(??, t) = u(?, t), ux(??, t) = ux(?, t) ? t>0, and the initial condition u(x, 0) = f(x), ?? ? x ? ?.
- Question : EX65 - (a) Verify that un(x, y) = exp ?ny ??n?sinnx, where n is a positive integer, is the solution of the Cauchy problem for the Laplace equation uxx + uyy = 0, x ? R, y > 0, u(x, 0) = 0, uy(x, 0) = n exp ???n?sinnx. (b) Show that this Cauchy problem is not well posed.
- Question : EX66 - Find the eigenvalues and eigenfunctions of the Sturm
- Question : EX67 - Show that the equation a2(x)u?? + a1(x)u? + ?a0(x) + ??u = 0,
- Question : EX68 - Reduce the given equation into the Sturm
- Question : EX69 - Determine the Euler load and the corresponding fundamental buckling mode of a simply supported beam of length a under an axial compressive force P which is governed by the eigenvalue problem
- Question : EX70 - Use the Fourier method to solve the Klein
- Question : EX71 - he Fourier method to solve the diffusion model ut = uxx + 2bux, 0 < x < a, t > 0, u(0, t) = 0 = u(a, t), t > 0, u(x, 0) = f(x), 0 < x < a
- Question : EX72 - (a) Solve the vibration problem of a circular membrane governed by (1.13.56) and (1.13.59), (1.13.60) with the boundary and initial conditions (1.13.52), (1.13.53) when g(r, ?) = 0.
- Question : EX73 - b) Obtain the solution for the problem (a) when f(r, ?) = a2 ? r2. 73. (a) Use the method of separation of variables to solve the Dirichlet problem in the cylinder for u(r, z) urr + 1 r ur + uzz = 0, 0 ? x ? a, 0 ? z ? h, u(r, 0) = 0 = u(a, z), u(r, h) = f(r). (b) If f(r) = 1, obtain the solution.
- Question : EX74 - (a) Derive the differential equality 2ut ?c2?2u ? utt? = 2c2?(utux)x + (utuy)y? ? ?c2?u2x + u2y? + u2t ?t, associated with the wave equation c2(uxx + uyy) ? utt ? c2?2u ? utt = 0. (b) Generalize the above differential equality for the (n + 1)-dimensional wave equation (1.13.9). (c) Show that the differential equality for the (n + 1)-dimensional wave equa- tion can be written in the form 2ut ?c2?2n ? utt? = 2c2?n
- Question : EX75 - (a) Derive the energy integral E(t) = 1 2 ? b a ?u2t + c2u2x + d2u2? dx, for the Klein
- Question : EX76 - (a) Use the method of separation of variables to solve the spherically symmetric wave equation (1.13.21) in three dimensions utt = c2?2u ? c2?urr + 2rur?, 0 < r < a, t > 0, u(a, t) = 0, t > 0, u(r, 0) = f(r), ut(r, 0) = g(r), 0 < r < a. (b) Find the solution for f(r) = 0 and g(r) = a ? r.
- Question : EX77 - Solve the axisymmetric Dirichlet problem in a right circular cylinder urr + r?1ur + uzz ? 0, 0 < r < a, 0 < z < h, u(r, 0) = 0 = u(r, h), u(a, z) = f(z).
- Question : EX78 - Use Example 1.6.4 to ?nd the solution of equation (1.6.80) with the boundary conditions (a) u(1, ?) = 2 cos2 ?, (b) u(1, ?) = |2?|, (c) ur(1, ?) = 2 cos 2?, (d) ur(1, ?) = cos ? + sin ?.
- Question : EX79 - Apply the method of separation of variables to solve the Laplace equation in an annular region urr + 1 r ur + 1 r2 u?? = 0, 0 < a < r < b, 0 ? ? ? 2? with the following boundary conditions: (a) u(a, ?) = f(?), u(b, ?) = g(?), 0 ? ? ? 2?. (b) u(1, ?) = 21 + sin ?, u(2, ?) = 21(1 ? 1n2 + 2 cos ?), 0 ? ? ? 2?.
- Question : EX80 - n Example 1.6.1, take c = 1, ? = ?. If u(x, t) = 21[f(x ? t) + f(x + t)] + ? x+t x?t g(?) d?, then show that ? ? 0 ?ux2 + ut2? dx = ?0??f?2 + g?2? dx, where f and g are real functions on 0 ? x ? ? with continuous partial deriva-
- Question : EX81 - Consider the boundary-value problem for the elliptic equation ? 2 u+u1==00 in D =?(x,y)?????|ax| + |by|?< 1?, on ?D, where a > b > 0. Show that a2b2 4(a2+b2) ? u(0,0)? a2 4
- Question : EX82 - Show that the Dirichlet problem ? 2 u = 0, x ? D, u(x) = f(x) on ?D, has a unique solution.
- Question : EX83 - Consider the Dirichlet boundary-value problem ? 2 u+u1==00, x??(x,y)?????|ax| + |by|?< 1?, on ?D, where a > b > 0. Use a suitable function v(x, y) = Ax2 + By2 satisfying ? 2v = 1 with (A, B) > (0, 0) to prove that (ab)2 2(a+b)2 ? u(0,0)? (ab)2 2(a2 + b2) .
- Question : EX84 - . Solve the fractional Blasius problem as stated in Exercise 35 with the governing equation (see Debnath 2003a, 2003b) ??u ?t? = ? ?2u ?z2 , x ? R, t > 0.
- Question : EX85 - Solve the fractional Stokes
- Question : EX86 - Apply the method of separation of variables u(x, t) = X(x)T (t) to solve the eigenvalue problem for the dissipation wave equation utt ? c2uxx + ?ut = 0, 0 < x < ?, t > 0, u(0, t) = 0 = u(?, t), t > 0, u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < ?. Show that (i) X?? + ?2X = 0, T
- Question : EX87 - SOLVE THE ABOVE PROBLEM 86 WITH THE SMALL INITIAL CONDITIONS AND THE FOLLOWING BOUNDARY CONDITIONS
- Question : EX88 - Solve the eigenvalue problem for the telegraph equation utt ? c2uxx + aut + b = 0, 0 < x < 1, t > 0, with the boundary and initial conditions u(0, t) = 0, ux(1, t) + u(1, t) = 0, t > 0, u(x, 0) = f(x) and ut(x, 0) = g(x), 0 < x < 1. Show that the eigenvalue equation and the eigenfunctions are tan ? = ? and Xn(x) = An sin ?x,
- Question : EX89 - Use the method of separation of variables to solve the problem utt ? c2uxx + ?u = 0, 0 < x < 1, t > 0, ux(0, t) = 0 = u(1, t), t > 0, u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < 1. Show that the eigenvalues and the eigenfunctions are ?n = ?41(2n ? 1)2?2c2 + ?? 1 2 , Xn(x) = An cos?(2n ? 1)?2x?, where n = 1, 2, 3, . . . , and ??2 is the separation constant. Derive the general solution u(x, t) = ? ? n=1 (an cos?nt + bn sin?nt)cos?(2n ? 1)?2x?. Show that the solution for f(x) = x and g(x) = 0 corresponds to an = 4 (2n ? 1)? ?(?1)n?1 ? 2 (2n ? 1)? ?, bn=0.
- Question : EX90 - Consider the eigenvalue problem utt ? c2uxx + au = 0, 0 < x < 1, t > 0, ux(0, t) = 0, ux(1, t) + u(1, t) = 0, t > 0, with the initial conditions u(x, 0) = f(x) and ut(x, 0) = g(x), 0 < x < 1, (a) Show that, for u(x, t) = X(x)T (t), c2X?? ? aX + ?2X = 0, T
- Question : EX91 - Consider the Helmholtz equation urr + 1 r ur + 1 r2 u?? + ?2u = 0 for r ? 1, where ? = ? c ? 1, with the boundary condition u(1, ?) = sin ?. Obtain the asymptotic solution in the form u = u0 + ?2u2 + O ??4?, where u = O(1) on the boundary. Show that the two-term asymptotic solution is u(r, ?) = r sin ? + 1 8 ?2 ?r ? r3? + O??4?.
- Question : EX92 - Consider the boundary value problem for the modi?ed Helmholtz equation ?2?2u = u, with u(1, ?) = 1 and u(r, ?) ? 0 as r ? ?. Show that the asymptotic solution is given by u=exp?1?r ? ?.
- Question : EX93 - (a) The temperature distribution u(x, t) in a homogeneous rod of length ? with insulated endpoints is described by the initial boundary-value problem ut = ?uxx, 0 < x < ?, t > 0, ux(0, t) = 0 = ux(?, t), t > 0, u(x, 0) = f(x), 0 < x < ?.
- Question : EX94 - (a) Show that the telegraph equation can be written in the form utt ? c2uxx + (p + q)ut + pqu = 0, where p = (G/C) and q = (R/L). (b) Apply the transformation u = v exp[?12(p + q)t] to transform the equation in the form vtt ? c2vxx = 1 4 (p ? q)2v. (c) When p = q, there exists an undistorted wave solution. Show that a pro- gressive wave of the form
- Question : EX95 - Consider the telegraph equation problem ut ? c2uxx + aut + bu = 0, 0 < x < l, t > 0, u(x, 0) = f(x), ut(x, 0) = g(x) for 0 < x < l, u(0, t) = 0 = u(l, t) for t ? 0, where a and b are constants. (a) Show that, for any T > 0 ? t 0 ?u2t + c2u2x + bu2?t=T dx ? ?0l?u2t + c2u2x + bu2?t=0 dx.
- Question : EX96 - Use the solution (1.9.15) to obtain the solution of the nonhomogeneous wave equation problem utt ? c2uxx = sin(kx ? ?t), x ? R, t > 0, u(x, 0) = 0 = ut(x, 0), x ? R. Discuss the solution for cases
- Question : EX97 - erive the Duhamel formula for the solution of Example 1.9.2 is u(x, t) = f(t) ? ? ?t u0(x,t)=? t 0 f(t? ?)???u?0?d? where u0(x,t) = L?1?1s exp??x?ks?? = erfc??4x?t?.
- Question : EX98 - Solve the axisymmetric unsteady viscous ?ow problem in a long rotating cylin- der of radius a governed by vt = ??vrr + 1rvr ? rv2?, 0 < r ? a, t > 0, where v = v(r, t) is the tangential ?uid velocity and ? is the kinematic viscos- ity of the ?uid.
- Question : EX99 - SHOW THAT THE SOLUTION OF THE CAUCHY PROBLEM FOR THE DIFFERENT EQUATION
- Question : EX100 - Verify that un(x, y) = exp ?ny ??n?sinnx is the solution of the Cauchy problem for the Laplace equation uxx + uyy = 0, x ? R, y > 0, u(x, 0) = 0, uy(x, 0) = n exp ???n?sinnx,
- Question : EX101 - (a) Verify that un(x,y) = 1 n e??n sinnxsinhny is the solution of the Cauchy problem for the Laplace equation in the upper half-strip uxx + uyy = 0, 0 ? x ? ?, y > 0, u(0, y) = 0 = u(?, y), y > 0 u(x, 0) = 0 and uy(x, 0) = e??n sin nx.
- Question : EX102 - Using the wave function ?(x, t) = a(x, t) exp[hi S(x, t)], where a and S are real functions and the transformation u(x, t) = m?1Sx in the Schr
- Question : EX103 - Consider the two-dimensional boundary value problem for the Laplace equa- tion in the upper half plane uxx + uyy = 0, x ? R, y > 0, u(x, 0) = f(x) and uy(x, 0) = g(x), x ? R. (a) If f(x) = 0 and g(x) = 0, show that u(x, y) ? 0 is the solution. (b) If the boundary data is changed to u(x, 0) = n1 cos nx and g(x) = 0, x ? R, show that u(x, y) = n1 cos nx cosh ny is the solution. Examine the ill-posedness of the problem.
- Question : EX104 - (a) Show that the solution of the Cauchy problem for the negative diffusion equation ut + ?uxx = 0, x ? R, t > 0, ? > 0, u(x, 0) = f(x), x ? R is given by u(x, t) = 1 n exp ??n2t?sinnx.
- Question : EX105 - (a) Show that the ill posed problem 104(a) can be made well posed by adding higher order diffusive terms, that is, the modi?ed Cauchy problem ut + ?uxx + ?uxxxx = 0, x ? R, t > 0 (?, ? > 0), u(x, 0) = 1 n sin nx, x ? R is a well posed problem. (b) Verify that u(x, t) = 1 n sin nx exp ??n2 ? ?n4?t is the solution of the modi?ed Cauchy problem.

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