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AME 60634 Int. Heat Trans. D. B. Go 1 sinh Function

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s inh Function. Fourier Series. Consider a set of eigenfunctions ϕ n that are orthogonal , where orthogonality is defined as. for m ≠ n. An arbitrary function f ( x ) can be expanded as series of these orthogonal eigenfunctions. or. Due to orthogonality , we thus know. - PowerPoint PPT Presentation

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AME 60634 Int. Heat Trans.

D. B. Go 1

sinh Function

AME 60634 Int. Heat Trans.

D. B. Go 2

Fourier SeriesConsider a set of eigenfunctions ϕn that are orthogonal, where orthogonality is defined as

for m ≠ n

An arbitrary function f(x) can be expanded as series of these orthogonal eigenfunctions

or

Due to orthogonality, we thus know

all other ϕnAmϕm integrate to zero because m ≠ n

Thus, the constants in the Fourier series are

AME 60634 Int. Heat Trans.

D. B. Go 3

Cartesian Sturm-LiouvilleCharacteristic Value Problem p(x) = 1; q(x) = 0; w(x) = 1

homogeneous B.C.

After Applying Final B.C. Typical B.C.

Dirichlet

Neumann

Robin

AME 60634 Int. Heat Trans.

D. B. Go 4

Cartesian Sturm-Liouville Kakac & YennerHeat Conduction, 3rd Ed.

AME 60634 Int. Heat Trans.

D. B. Go 5

Cylindrical Sturm-LiouvilleCharacteristic Value Problem p(r) =r; q(r) = −ν2/r; w(r) = r

homogeneous B.C.

After Applying Final B.C. Typical B.C.

Dirichlet

Neumann

Robin

AME 60634 Int. Heat Trans.

D. B. Go 6

Cylindrical Sturm-Liouville

homogeneous B.C.

Typical B.C.

Dirichlet

Neumann

Robin

Special B.C. case: a = 0, b = r0

After Applying Final B.C.

AME 60634 Int. Heat Trans.

D. B. Go 7

Cylindrical Sturm-Liouville Kakac & YennerHeat Conduction, 3rd Ed.