MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 4 pptx

... equations in physics and engineering can be solved by the method of separation of variables. This method helps us to reduce a second-order partial differential equation into a set of ordinary ... following integral definitions: (6 .48 ) and 1 2 (1 - t2)n-6 cosztdt, (n > ). (6 .49 ) Jn(IL.1 = 6 .4 RECURSION RELATIONS OF THE BESSEL FUNCTIONS Using the s...

Ngày tải lên: 13/08/2014, 09:21

... by determining the u and the solutions in Equation (19.15) as sin bx and sin ko(x - L), respectively. We can now obtain a closed expression for G(z, d) as sin kox sin ICg(x’ ... dk'g(k')eikfZ VG -" and Their inverse Fourier transforms are and Using these in Equation (19.116) we get which gives us Substituting this in Equation (19.118) we...

Ngày tải lên: 13/08/2014, 09:21

... 14. 7 Applications of Diflerintegrals in Science and Engineering 42 4 14. 7.2 Fractional Fokker-Planck Equations 42 7 Problems 42 9 14. 7.1 Continuous Time Random Walk (CTRW) 42 4 15 INFINITE ... 44 0 15.5 Useful Inequalities About Series 44 2 15.6 Series of Functions 44 2 15.6.1 Uniform Convergence 44 3 15.6.2 Weierstrass M-Test 44 3 15.6.3 Abel Test 44 4...

Ngày tải lên: 13/08/2014, 09:21

... part of the learning process. In a vast area like mathematical methods in science and engineering, there is always room for new approaches, new applications, and new topics. In fact, the number ... write x = fl in the generating function Equation (2.65) we find (2.85) Expanding the left-hand side by using the binomial formula and comparing equal powers oft, w...

Ngày tải lên: 13/08/2014, 09:21

... as (4. 53) and defining a new parameter (4. 54) 50 - = t, 2 and using the generating function [Eq. (4. 25)], we obtain (4. 55) We now use the orthogonality relation [Eq. (4. 46)] ... (z) , (n 2 m). (4. 41) J-00 Using the Rodriguez formula, we write Equation (4. 41) as After n-fold integration by parts and since n > m, we obtain (4. 43) Since...

Ngày tải lên: 13/08/2014, 09:21

... exists a minimum value, mmin, thus determining X as X = mmin (9. 143 ) To find mmin we equate the two expressions [Eqs. (9. 141 ) and (9. 143 )] for X to obtain (9. 144 ) mmin = -1. ... Substituting this into Equation (9.83) and integrating gives us ko(z) = cacota(z +p) + . d sin+ + p) ’ (9.89) where p and d are integration constants. With thes...

Ngày tải lên: 13/08/2014, 09:21

... 1 0 - sin $ cosCp sin4cos$ cosdsin+ -sin4 cos~cos$ (10.93) Reversing the order we get cos $ (10. 94) sin $ sin Cp - sin $ cos Cp R2R1 = 0 cos Cp sin 4 [ sin $ - cos ... choosing three mutually orthogonal straight lines. A point is defined by giving its coordinates, (q,z2,q), or by using the position vector 163 188 COORDINATES AND TENSORS Hence aa...

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... be expressed in terms of the poten- tials -A' and 4 as In the Lorentz gauge 2 and 4 satisfy and (10.373) (10.3 74) (10.375) (10.376) (10.377) respectively. Defining a four-potential ... transformation. It gives i3, as a linear Ah' combination of Gar. Using we obtain (10. 340 ) (10. 341 ) (10. 342 ) 222 COORDINATES AND TENSORS and 10.2...

Ngày tải lên: 13/08/2014, 09:21

... (Q 74) Km(Q, 4) = 4P fimmf and the completeness relation (1 1. 240 ) ( 1 1. 24 1) c c Y,L(Q’ ,4 )Km(B ,4) = G(cosQ-cosQ’)6 (4- 4’), I=O m=-I (11. 242 ) where dQ = sin Qdsdq5. In the ... 256 CONTINUOUS GROUPS AND REPRESENTATIONS and (11. 244 ) aa Glrn(e ,4) = o(- -,Q ,4) Krn(Q ,4) , a6 ' a4 where O($, &, 8 ,4) is some differential operator...

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... (11 .44 1) (11 .44 2) (11 .44 3) (11 .44 4) 12 COMPLEX VARIABLES and FUNCTIONS Even though the complex numbers do not exist directly in nature, they are very useful in physics and engineering applications: ... dimensions. 2. The method of analytic continuation is very useful in finding solutions of differential equations and evaluating some definite integrals....

Ngày tải lên: 13/08/2014, 09:21