By Mircea Soare, Petre P. Teodorescu, Ileana Toma
The current publication has its resource within the authors' desire to create a bridge among the mathematical and the technical disciplines, which desire a strong wisdom of a robust mathematical instrument. the need of such an interdisciplinary paintings drove the authors to put up a primary publication to this goal with Editura Tehnica, Bucharest, Romania.The current e-book is a brand new, English version of the amount released in 1999. It comprises many advancements about the theoretical (mathematical) details, in addition to new subject matters, utilizing enlarged and up to date references. purely usual differential equations and their strategies in an analytical body have been thought of, leaving apart their numerical approach.The challenge is first of all acknowledged in its mechanical body. Then the mathematical version is decided up, emphasizing at the one hand the actual value taking part in the a part of the unknown functionality and nonetheless the legislation of mechanics that result in a standard differential equation or process. the answer is then bought by way of specifying the mathematical tools defined within the corresponding theoretical presentation. ultimately a mechanical interpretation of the answer is supplied, this giving upward thrust to an entire wisdom of the studied phenomenon.The variety of functions used to be elevated, and plenty of of those difficulties look at present in engineering.
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Extra resources for Ordinary Differential Equations with Applications to Mechanics
Sample text
Let now y = y j j =1,n , f = f j j =1,n be vector functions and assume that we must solve [ ] [ ] the vector equation Ly ≡ &y& + p (x )y& + q(x )y = f , ( ) p, q ∈ C 0 (I ), f ∈ C 0 (I ) . 30) componentwisely, this means, in fact, that one has to solve n uncoupled ODEs Ly j ≡ &y& j + p (x ) y& j + q (x ) y j = f j , j = 1, n . 40), y j (x ) = k j1Y1 (x ) + k j 2 Y2 (x ) + Y2 (x )∫ Y1 (x ) f j (x ) W (x ) dx −Y1 (x )∫ Y2 (x ) f j (x ) W (x ) dx, j = 1, n. 48), written in vector form, is Y1 (x )f (x ) Y (x )f (x ) dx −Y1 (x ) 2 dx, W (x ) W (x ) , k 2 = k j 2 j =1,n .
Some of the polynomials obtained this way have various applications. Thus, 1 1 2 n n! ⎛ ⎞ F ⎜ − n, n + , ; x 2 ⎟ = (− 1)n P2 n (x ), 2 2 1 ⋅ 3 ⋅ 5 K (2n − 1) ⎝ ⎠ 3 3 2 n n! ⎛ ⎞ xF ⎜ − n, n + , ; x 2 ⎟ = (− 1)n P2 n +1 (x ), 2 2 1 ⋅ 3 ⋅ 5 K (2n + 1) ⎝ ⎠ . 126) where P j (x ) are Legendre’s polynomials, satisfying the equation (x −1)y ′′ + 2xy ′ − n(n + 1)y = 0 . 127) Jacobi’s polynomials, more general than Legendre’s, are obtained by considering Q n (x ) ≡ F (n,− n + α, β; x ) = [ ] x 1−β (1 − x )β −α d n β + n −1 (1 − x )α + n −β .
The tangent at the origin is ε& = σ 0 / η . The time-dependent function ϕ(t ) = 1 − e E − t η is called the creep function. 5 Problem. Determine the general meridian displacements w of a thin shell of rotation. Particular case: the spherical dome of radius a, acted upon by its own weight g. Mathematical model. g. Flügge) dw − w cot ϕ = f (ϕ) , dϕ (a) where φ is the angular variable (the meridian angle) and f (ϕ) is a function depending on the external loading. Solution. 3. 2, the general solution whomog = C sin ϕ .