Basic Concepts 1. Hyperbolic Equations[1] A partial differential equation of second order, i.e., one of the form Auxx + 2Buxy + Cuyy + Dux + Euy +F = 0 is called hyperbolic if B 2 - 4AC < 0 Example: The one dimensional wave equation: utt - c2 uxx = 0 is an example of a hyperbolic equation. 2. Riemann Function[2] The Riemann function R(x, y; ζ, η) is defined as the solution of the equation Rxy - (aR)x - (bR)y + cR = 0 which satisfies the conditions y
Z R(ζ, y; ζ, η) = exp[
a(ζ, t) dt]
η
and Z R(x, η; ζ, η) = exp[
x
b(t, η) dt]
ζ
on the characteristics x = ζ and y = η, where (ζ, η) is a point on the domain Ω on which the conditions state above are defined. The solution is then given by the Riemann formula Z u(x, y) =
x
Z dζ
0
y
R(ζ, η; x, y)f (ζ, η) dη 1
This method of solution is called the Riemann method. 3. Green’s theorem[3] Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then Z ZZ ∂L ∂M − ) dx dy (L dx + M dy) = ( ∂y C D ∂x where the path of integration along C is counterclockwise. 4. Adjoint Equation[4] The adjoint equation is applicable to linear differential equations which yield a linear differential equation of a lower order. For every solution of the adjoint equation we can find, we can reduce the order of the original equation by one. 1
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5. Adjoint Linear Operators[5] To a linear operator L, defined by L(u) = auxx + 2huxy + buyy + 2gux + 2f uy + cu where the coefficients are continuously differentiable functions of x and y alone, there corresponds a unique linear operator L∗ , called the adjoint of L, such that vL(u) - uL∗ (v) is a divergence. If D is a domain whose boundary is a regular closed curve C, it follows by Green’s theorem that ZZ Z (vL(u) − uL∗ (v)) dx dy = (lH + mK) ds D
C
where (l, m) are direction cosines of the outward normal to C. After doing some calculations and subsequent simplifications, we get, L∗ (v) = (av)xx + 2(hv)xy + (bv)yy − 2(gv)x − 2(f v)y + cv where H = avux − u(av)x + hvuy − u(hv)y + 2guv K = hvux − (hv)x + bvuy − u(bv)y + 2f uv H and K are not unique; we can add to them ∂θ/∂y and -∂θ/∂x respectively; but L∗ is unique and the adjoint of L∗ is L. L is said to be self-adjoint if L∗ = L. The condition for this is ax + hy = 2g, hx + by = 2f so that a self-adjoint operator can be written in the form ∂ ∂ ∂ ∂ (aux ) + (huy ) + (hux ) + (buy ) + cu ∂x ∂x ∂y ∂y
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Riemann Method 6. Use[4] The Riemann’s method is applicable to linear hyperbolic equations of the second order in two independent variables which yields an exact solution in terms of the solution to the adjoint equation. 7. Idea[4] The solution of a non-characteristic initial value problem in two dimensions can be found if the adjoint equation with specific boundary conditions can be solved. 8. Essentials of Riemann Method[6] The solution of the problem of Cauchy given by Riemann for the partial differential equation (1) uxy - a(ux − uy ) = 0, a = a(x, y) has been extended to the general linear partial differential equation of second order in two independent variables x, y, (2) uxy + aux + buy + cu = f where a, b, c, f are given functions of x, y. For the sake of simplicity we consider the equations of the type (3) L(u) = uxy - aux - buy = 0, a = a(x, y), b = b(x, y) Actually, if a solution u1 of (2) and a solution u0 of the homogeneous equation (f = 0) are known, equation (2) can always be reduced to one of type (3) for a new unknown function u defined by u = u1 + u0 u. The classical procedure associates with (3) its adjoint equation (4) M (v) = vxy + (av)x + (bv)y = 0 and is based on the differential identity (5) Ax + By = vL(u) - uM (v), where A =
(vuy −uvy ) 2
- auv, and B =
(vux −uvx ) 2
- buv
The identity implies that the line integral Z I = (B dx − A dy) vanishes around closed paths lying in the interior of a domain D within which the function u, v are regular solutions of L(u) = 0 and M (v) = 0, respectively. The line integral I vanishes around closed paths leads immediately to the solution of Cauchy’s problem for (3). Let C be an arc meeting any horizontal or vertical line in at most one point. Along C the values of ux , uy (Cauchy’s data) for the
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solution of (3) are given. In order to calculate the value u(P ) of the solution at the point P in the figure, the line integral I is evaluated around the closed path P XY P formed by the horizontal line segment P X, the arc XY of C, and the vertical line segment Y P . One finds (6) RY RX RP B(x, y) dx + X (B dx − A dy) - Y A(x, y) dy = 0 P If a solution v = v(x, y; x, y) of M (v) = 0 can be found for which (7) vx = -bv on y = y, vy = -av on x = x and v(x, y; x, y) = 1 hold on the characteristics x = x and y = y through P , we shall have (8) A(x, y) = (uv)y /2 and B(x, y) = (uv)x /2 and consequently (6) will yield (9) RY u(P ) = 21 [u(X)v(X) + u(Y )v(Y )] + X (B dx − A dy) The function v is known as Riemann’s function, and, once it has been determined, u(P ) may be calculated from (9) from the knowledge of Cauchy data along C. The first step in Riemann’s method is the formulation of the line integral I so that it vanishes around closed paths. We note that the coefficients A, B of the differentials are bilinear forms in two set of variables u, ux , uy and v, vx , vy . Furthermore, the real reason for the introduction of adjoint equation (4) and its solution v is to make available the identity (5), which guarantees that I vanishes around closed paths. 9. A more general case[5] Consider the following equation L(u) = 2uxy + 2gux + 2f uy + cu = F (x, y) given u, ux , uy on some duly rectangular arc C. The data satisfy the strip condition du = ux dx + uy dy on C. The functions g, f , c, F are assumed to be continuously differentiable functions of x and y alone. Let the characteristics y = y0 and x = x0 through P (x0 , y0 ) cut C in Q and R respectively. Let D be the domain bounded by P Q, P R and C; let Γ be its boundary. Then, if L(u) = F , L∗ (v) = 0, we have
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ZZ
ZZ
(vL(u) − uL∗ (v)) dx dy =
vF dx dy = D
D
ZZ [ D
∂H ∂K + ] dx dy = ∂X ∂y
Z (−K dx+H dy) Γ
where H = avux - u(av)x + hvuy - u(hv)y + 2guv and K = hvux - (hv)x + bvuy - u(bv)y + 2f uv by Green’s theorem, assuming that u and v have continuous second derivatives. Using the aforementioned values of H and K, the integral round Γ is equal to Z Z Z − [(vu)x − 2u(vx − f v)] dx + (−K du + H dy) + [(vu)y − 2u(vy − gv)] dy PQ QR RP Z = 2(uv)P − (uv)Q − (uv)R + (−K dx + H dy) QR
provided that vx − f v = 0 when y = y0 vy − gv = 0 when x = x0 that is, provided that Z
x
v(x, y0 ) = exp
f (ξ, y0 ) dξ x0
and Z
y
v(x0 , y) = exp
g(x0 , η) dη y0
where we have taken v(x0 , y0 ) to be unity, as we may without the loss of generality. Hence v is given by a characteristic boundary value problem for L∗ (v) = 0. This problem has a solution, the Riemann-Green function, which we denote by v(x, y; x0 , y0 ). If Q is (x1 , y0 ), and R is (x0 , y1 ), it follows that 1 1 u(x0 , y0 ) = u(x1 , y0 )v(x1 , y0 ; x0 , y0 ) + u(x0 , y1 )v(x0 , y1 ; x0 , y0 ) 2 2 Z ZZ 1 1 (K dx − H dy) + v(x, y; x0 , y0 )F (x, y) dx dy 2 QR 2 D
+
This is the required solution, since H = vuy - uvy + 2guv and K = uvx - vux + 2f uv are given on QR by the Cauchy data. 10. Determination of the Riemann-Green function[5] The difficulty in Riemann’s solution is the determination of the Riemann-Green function. The method replaces a Cauchy problem by a characteristic boundary value problem. It suffices to consider the case when independent variables are characteristic variables, so that the linear operator is L(u) = 2uxy + 2gux + 2f uy + c
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If we make the change of dependent variable u = φU , and divide through by φ, we get a linear operator M (U ) = 2Uxy + 2GUx + 2F Uy + CU where G=g+
φy φ
F =f+
φx φ
and
and C = c + 2g
φx φy φxy + 2f + φ φ φ
The Riemann-Green function for L(u), v(x, y; x0 , y0 ), satisfies the adjoint equation L∗ (u) = 0 and the conditions vx = f v on y = y0 , vx = gv on x = x0 and v(x0 , y0 ; x0 , y0 ) = 1 It follows that the Riemann-Green function for M (U ) = 0 is V (x, y; x0 , y0 ) =
φ(x, y) v(x, y; x0 , y0 ) φ(x0 , y0 )
For example, if we put u = (x + y)U in uxy = 0, for which the Riemann-Green function is constant, we get Uxy +
Ux + Uy =0 x+y
for which the Riemann-Green function is therefore V (x, y; x0 , y0 ) =
x+y x0 + y0
The change of dependent variable is useful if we can choose φ so that M (U ) = 0 is self-adjoint. This occurs if ∂ ∂ logφ, g = - ∂y logφ f = - ∂x
such a transformation is thus possible when gx = fy . The Riemann-Green function is the solution of a characteristic boundary value problem, and does not depend in any way on the arc carrying the Cauchy data. If it is possible to solve by some other method the problem of Cauchy with a simple curve carrying the data, a comparison of this solution and the Riemann solution should give the Riemann-Green function. In the case of the two equations discussed by Riemann, it was possible to do this; he solved the problem of Cauchy with data on a straight line by using Fourier cosine transforms.
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11. An Example[4] Consider the following partial differential equation (E1) α2 wββ − β 2 wαα = 0 w(α, 1) = f (α) wβ (α, 1) = g(α) where -∞ < α < ∞, 1 < β < ∞, and f (α) and g(α) are given functions. If we change variables in equation (E1) from {w, α, β} to {u, x, y} by u(x, y) = w(α, β) β x = αβ and y = α Now, equation (E1) becomes upon doing the transformations stated above (10) 1 uxy - 2x uy = 0
The boundary conditions in equation (E1) transform to (11) u(s, 1s ) = f (s) and sux (s, 1s ) + 1s uy (s, 1s ) = g(s) where −∞ < s < ∞. By manipulations of equation (11), we can derive the following set of equations (E2), 1 u(s, ) = f (s) s 1 1 1 ux (s, ) = [f 0 (s) + g(s)] s 2 s 1 1 uy (s, ) = [sg(s) − s2 f (s)] s 2 The domain in which equations (10) and (E2) are to be solved is shown in this figure.
To solve equations (10) and (E2), we use Riemann’s method. Comparing equation (10) to L[u] = uxy + a(x, y)ux + b(x, y)uy + c(x, y)u = f (x, y), we determine a = 0, b = −1/2x, c = 0, f = 0 The Riemann’s method gives solution of the form Z Q ZZ 1 1 u(ζ, η) = R(P ; ζ, η)u(P ) + R(Q; ζ, η)u(Q) − B[u(x, y), R(x, y; ζ, η)] + f (x, y)R(x, y; ζ, η) dx dy 2 2 P D This formula is valid when we wish to find u(S) = u(ζ, η), where S represents an arbitrary point as shown in the figure below. If we assume that the initial curve Γ is monotonically decreasing then we can write the solution in the above stated
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form. Here, P and Q are as shown in the figure below.
Coming back to our problem at hand (10), we can write the solution (E3) as follows Z Q 1 1 1 1 1 1 1 u(ζ, η) = R(P ; ζ, η)u(P ) + R(Q; ζ, η)u(Q) − [( Ruy − Ry u) dy − (− Ru + Rux − Rx u) dx] 2 2 2 2 2x 2 2 P All that remains to find the Riemann’s function. Using equations from section 2, we say that R(x, y; ζ, η) satisfies (E4) 1 Ry = 0 Rxy + 2x R(ζ, y; ζ, η) = 1 r ζ R(x, y; ζ, η) = x R(ζ, η; ζ, η) = 1 Because the first equation of (E4) can be integrated directly with respect to x and then with respect to y the general solution to equation (E4) is easily seen to be of the form (12) √ R(x, y; ζ, η) = M (x; ζ, η) + K(y;ζ,η) x for some M (x; ζ, η) and some K(y; ζ, η). Using equation (12) in the boundary conditions in the set of equations (E4), the solution is found to be (13) q R(x, y; ζ, η) =
ζ x
Using equation (13) in equation (E3), we can find u(ζ, η) and hence, w(α, β) for any values of α and β. 12. Riemann Zeta Function[7] The Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of infinite series ∞ X 1 n=1
ns
which converges when the real part of s is greater that 1. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Definition: The Riemann zeta function ζ(s) is a function of a complex variable s = σ + ιt.
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The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case: ∞ X 1 ζ(s) = ns n=1 where σ = R(s) > 1 It can also be defined by the integral Z ∞ s−1 1 x ζ(s) = dx Γ(s) 0 ex − 1 The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of preceding series.
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References [1] [2] [3] [4] [5] [6] [7]
Wolfram Mathworld: Hyperbolic Partial Differential Equations Wolfram Mathworld: Riemann Function Wikipedia: Green’s Theorem Zwillinger D. (1997), Handbook of Differential Equations 3rd edition, Academic Press Copson E.T., Partial Differential Equations, Cambridge University Press 1975 Martin M.H., Riemann’s Method and the Problem of Cauchy, University of Maryland Wikipedia: Riemann Zeta Function
