User:Egm6322.s12.team2.steele.m2/Mtg4

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg4.djvu|Mtg 4]] Tue, 30 Aug 11
=Simplify for Analytical Solution=

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=Simplify for Analytical Solution= Complex 		 Simplified 2nd order		=	2nd order Nonlinear		-> 	linear Unknown varying	->	known varying coefficients Coefficients Note: The formulas for the coefficients $$c_i (Y^1,t)$$ for i=0, 1, 2, 3 are known, but not their values until $$u^1$$ and $$u^2$$ are known (i.e., solved for). General structure of Linear 2nd-order ODEs with varying coefficients (L2-ODE-VC)

X = independent variables Y(x) = dependent variable (unknown function to be solved for)

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Ref: King 2003, Differential equations, p.3. Caveat: “… “exact” solutions of a differential equation … Winsberg appears to believe that numerically evaluating the closed-form solutions to the equation provides the best results. However, it’s well known that for some functions, including the Airy functions and the Hankel functions, approximating the differential equation is more numerically precise than evaluating the closed-form function directly. In general, closed-form solutions won’t give more insight than will the differential equation itself.” F. Sullivan, Reviewer of Science in the Age of Computer Simulation by E. Winsberg 2010, Physics Today, Aug. 2011, p. 50. See, e.g., SPECFUN – Special Function Evaluation http://people.sc.fsu.edu/~jburkardt/f_src/specfun/specfun.html

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Motivation: Many linear PDEs in engineering applications can be solved by separation of variables. Some examples include, but are not limited to: Heat, solids, fluids, acoustics and electromagnetic. Examples of these PDEs are:

Ref: PEA1 F09 Mtgs 28, 29, 30 A point x in 3-D has Cartesian coordinates $$(x_1, x_2, x_3)$$ and is written as

Consider the separation of variables $$X(x) = X_1(x_1) X_2(x_2) X_3(x_3)$$ $$x = (x_1, x_2, x_3)$$

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See Cartesian curve for $$x = (x_1, x_2, x_3)$$. See Curvilinear curve for $$\xi = (\xi_1, \xi_2, \xi_3)$$. A point $$\xi$$ in 3-D has curvilinear coordinates $$(\xi_1, \xi_2, \xi_3)$$ and is written as

Consider the separation of variables:

Separated equations for i=1,2,3

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R1.4: Draw the polar coordinate lines, in a 2-D plane emanating from a point, not at the origin. Change symbols to simplify: $$\xi_i \rightarrow x$$ $$X_i(\xi_i) \rightarrow y(x)$$

$$f_i(\xi_i) \rightarrow a_0(x)$$ R1.5: Show that (3) p.4-4 becomes

Note that $$a_1(x) = \frac{g^\prime (x)}{g(x)}$$ Number (2) is a particular case of (1) p.4-1, which in turn can be written as

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=Definition: Linearity= Consider an operator $$F(\cdot)$$ is linear if and only if (iff)

Note: $$\forall$$ = forall $$\in$$ = in, belonging to 	$$\exists$$ = there exists $$\mathbb R$$ = set of all real numbers (bb = blackboard for \mathbb)