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Analytical Methods (2 sessions) . Separation of Variables Method. Separation of variables sometimes called the method of Fourier for solving a PDE: Separation of variables in rectangular coordinates for Laplace’s Equations: Dirichlet boundary conditions:
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Separation of Variables Method • Separation of variablessometimes called the method of Fourier for solving a PDE: • Separation of variables in rectangular coordinates for Laplace’s Equations: • Dirichlet boundary conditions: • After applying boundary condition: • Using superposition:
Separation of Variables Method • Separation of variables in rectangular coordinates for wave equation: • After applying boundary condition:
Separation of Variables Method • Another Example: • Method of separation of variables for a problem with four inhomogeneousboundary conditions: • Using superposition: = + + +
Separation of Variables Method • Separation of Variables in Cylindrical Coordinates: • Laplace’s Equation in Cylindrical Coordinates : • After ... We have: • by the superposition principle to form a complete series solution:
Separation of Variables Method • Wave Equation in Cylindrical Coordinates: • Dividing both sides by ρ2: Using: Using: is Bessel’s equation Assuming x=λρand R=y
Separation of Variables Method • Jn & Ynarefirst and second kinds of ordern • Ynis also called the Neumann function • Bessel’s equation has a general solution: • We may replace xby jx, modified Bessel’s equation is generated: • ModifiedBessel’s equation has a general solution: • In & Knarefirst and second kinds of ordern
Separation of Variables Method Gamma function • Final solution is:
Separation of Variables Method • Asymptotic expressions:
Separation of Variables Method • Example: • A plane wave E=azEo e−jkxis incident on an infinitely long conducting cylinder: • Determine the scattered field. • Solution: • Integrating over 0≤φ≤2π gives: • Taking the m-thderivative of both sides: • Since Eszmust consist of outgoing waves that vanish at infinity, it contains: • Hence:
Separation of Variables Method • Solution (cont.): • The total field in medium 2 is: • While the total field in medium 1 is zero, At ρ=a, the boundary condition requires that the tangential components of E1 and E2 be equal. Hence:
Separation of Variables in Spherical Coordinates • Laplace’s equation for finding potential due to an uncharged conducting spherelocated in an external uniform electric field: • To solve (1): • To solve (2): (1) is Cauchy-Euler equation (2)
Separation of Variables in Spherical Coordinates Legendre differential equation • Making these substitutions: • Its solution is obtained by the method of Frobenius as: Legendre functions of the first and second kind
Separation of Variables in Spherical Coordinates • Qnare not useful since they are singular at θ=0,π • We use Qnin problems having conical boundaries that exclude the axisfrom the solution region. • For this problem, θ=0,πis included so that: • After determining Anand Bn, using boundary conditions:
Separation of Variables in Spherical Coordinates • Wave Equation: • is one of: • is one of: spherical Bessel functions Using: ordinary Bessel functions
Separation of Variables in Spherical Coordinates • By replacing H=y & cosθ=x, thesecond equation, which is Legendre’s associated differential equation, can be rewritten: • General solution is: • Using ordinary Legendre functions: Legendre functions of the first and second kind
Separation of Variables in Spherical Coordinates • Example: A thin ring of radius a carries charge of density ρ. • Find V(P(r, θ,φ) at: (a)point P(0, 0, z) on the axis of the ring, (b) point P(r, θ,φ) in space. • Solution: From elementary electrostatics, at P(0, 0, z): • Using ∇2V=0 havingboundary-value solution in P(0, 0, z): • Since Qnis singular at θ=0,π,Bn=0, Thus: • For 0≤r≤a, Dn=0 since V must be finite at r=0: • Using boundary-value:
Separation of Variables in Spherical Coordinates • Solution (cont.): • For r≥a, Cn=0since V must be finite as r→∞: • Therefore: ?
Separation of Variables in Spherical Coordinates • Example: • A PEC spherical shell of radius is maintained at potential Vocos2φ • Determine Vat any point inside the sphere. • Solution: • Solution is similar to that of previous problem • Using the boundary condition:
Orthogonal Functions • Orthogonal functions, including Bessel, Legendre, Hermite, Laguerre, and Chebyshev, are useful in solution of PDE. • They are very useful in series expansion of functions such as Fourier-Bessel series, Legendre series, etc. • Besseland Legendre functions are importance in EM problems. • A system of real functions φn(n=0,1,2,…) is said to be orthogonal with weight w(x) on (a, b) if: • For example, the system of functions cos(nx)is orthogonal:
Orthogonal Functions • An arbitrary function f(x), defined over interval (a, b), can be expressed in terms of any complete, orthogonal set of functions: Where: Others are in Table 2.5
Orthogonal Functions • Example: • Expand f(x) in a series of Chebyshevpolynomials. • Let: • Since f(x)is an even function, odd terms in expansion vanish: • By multiply both sides by: • and integrate over −1≤ x≤1:
Series Expansion • As noticed, PDE can be solved with the aid of infinite series and generally, with series of orthogonal functions. • An example: • Let the solution be of the form: • by substituting this equation into PDE:
Practical Applications Using: • Scattering by Dielectric Sphere: • It is illuminated by a plane wave propagating in z direction and Epolarized in xdirection. • As a similar way: … … = =
Practical Applications • Scattering by Dielectric Sphere (cont.): • Form of the incident fields: • Form of the scattered fields:
Practical Applications • Scattering by Dielectric Sphere (cont.): • Similarly, transmitted field inside the sphere: • Continuity of tangential components at surface: • Radar Cross Section (RCS): • By using asymptotic expression (R=∞) for spherical Bessel functions: • Where amplitude functions S1(θ)and S2(θ)are given as:
Practical Applications • Scattering by Dielectric Sphere (cont.): • Using: • Similarly, forward-scattering cross section:
Practical Applications • Attenuation Due to Raindrops: • At f>10GHz, attenuation caused by atmospheric particles (oxygen, ice crystals, rain, fog, and snow) can reduce the reliability and performance of radar and space communication links. • We will examine propagating through rain drops: • By Mie solution, the attenuation and phase shift of an EM wave by raindrops having spherical shape (for low rate intensity) is investigated. • For high rain intensity, an oblate spheroidal model would be more realistic as: • EM attenuation due to travel a wave through a homogeneous medium (with Nidentical spherical particles/m3) in a distance is given by e−γ. • γ is attenuation coefficient as [11]: • To relate attenuation and phase shift to a realistic rainfall, drop-size distribution (D) for a given rate intensity N(D) must be known. • Total attenuation and phase shift over the entire volume: [11] H.C. Van de Hulst, “Light Scattering of Small Particles”. New York: JohnWiley,1957, pp. 28–37, 114–136, 284.
Practical Applications Laws and parsons drop-size distributions for various rain rates • Attenuation Due to Raindrops (cont.): Raindrop terminal velocity
Numerical Integrations • Numerical integration is used whenever a function cannot easily be integrated in closed form. • The common ones include: • Euler’s rule (Riemann sum), • Trapezoidal rule, • Simpson’s rule, • Newton-Cotes rules, and • Gaussian (quadrature) rules. • Euler’s Rule (Rectangular Rule) (Riemann Sum): • We divide the curve into n equal intervals as shown: • This method gives an inaccurate result since each Ai is less or greater than the true area introducing negative or positive error, respectively.
Numerical Integrations • Euler’s Rule (cont.): • Other forms:
Numerical Integrations • Trapezoidal Rule:
Numerical Integrations • Simpson’s Rule: • Simpson’s rule gives more accurate result than the trapezoidal rule. • While trapezoidal approximates curve by connecting successive points on curve by straight lines, Simpson’s rule connects successive groups of three points on curve by a second-degree polynomial (i.e., a parabola). • where n is even. • Computational molecules for integration:
Numerical Integrations Where • Newton-Cotes Rules: • To apply, we divide interval a<x<b into m equal intervals so that: • Where m is a multiple of n, and n is number of intervals covered at a time or order of approximating polynomial. • Subarea in the interval xn(i−1)<x<xni: • Integration is: • Most widely known Newton-Cotes formulas are: • n=1 (2-point; trapezoidal rule): • n=2(3-point; Simpson’s 1/3 rule): • n=3(4-point; Newton’s rule): Newton-Cotes Numbers
Numerical Integrations • Gaussian Rules: • Idea of integration rules using unequally spaced abscissa points stems from Gauss. • Gaussian rules are more complicated but more accurate than Newton-Cotes. • A Gaussian rule: • i is a sequence of orthogonal polynomials and xiare zeros of i. • Any of orthogonal polynomials discussed in CH2can be used to give a particular Gaussian rule. • Commonly used rules are: • Gauss-Legendre, • Gauss-Chebyshev, etc., • Points xi are the roots of the Legendre, Chebyshev, etc., of degree of n. • For Legendre(n=1 to 16) and Laguerre (n=1 to 16) polynomials, the zeros xiand weights wihave been tabulated in the paper:
Numerical Integrations where • Using Gauss-Legendrerule: • Gauss-Chebyshev rule is similar to Gauss-Legendre rule. • We use above eq. except that sample points xi, roots of Chebyshev polynomial Tn(x), are: • The weights are all equal: • When limits of integration are ±∞: • A major drawback is that to improve accuracy, n must be increased which means that wi & ximust be included in program for each value of n. • Another disadvantage is that function f(x) must be explicit since sample points xiare unassigned. are transformation of roots xiof Legendre polynomials Abscissas (Roots of Legendre Polynomials) and Weights for Gauss-Legendre Integration
Numerical Integrations • Multiple Integration: • This is an extension of one-dimensional (1D) integration. • Substitution: • Procedure can be extended to a 3D integral:
Numerical Integrations • Integration Molecule: Triple integration molecule for Simpson’s 1/3 rule Double integration molecule for Simpson’s 1/3 rule