# Questions tagged [variational-calculus]

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### How to solve calculus of variations problems numerically?

For example, how to solve the well-known isoperimetric problem (i.e., to enclose the largest area with a fixed-length curve)? We can simplify things a bit and fix the two ends of the curve at $[a,0]$,...
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### Weak form of the Navier-Cauchy equation

I am trying to obtain the weak form of the Navier-Cauchy equation, which is $$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$ ...
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I am confused. I read an introduction to finite element method where it was derived for the poisson equation: $$-\Delta u + cu = f,\qquad, u = g_0 \text{ on Dirichlet boundaries},\qquad\partial_n u ... 0answers 147 views ### Best way to compute given functional with accuracy: I need to plot the following functional with accuracy:$$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1}, $$Where  F(z,s) = \dfrac{1}{z^s\Gamma(\... 1answer 71 views ### Prove that the set of maximizers are independent of parameter in the objective function A maximization problem reads as$$ J(y) = \sum_{k=1}^{K} \sigma_k(y)^q \mathop{\rightarrow}^{y} max$$where q \in [1,\infty] is a user-defined parameter and functions \sigma_k, k=\{1,\dots,K\} ... 0answers 106 views ### functional second derivative I'm trying to build a numerical solution for a parameter estimation problem of reaction-diffusion equation, using the adjoint method. To summarize it, I'm trying to minimize the function$$ G=\frac{...
Calculus of variations problems are generally cast in in the following simple form: find $u(t)$ that satisfies some boundary conditions and minimises  F[u] = \int_{t=0}^{t=t_f} f(u(t),u'(t),t) dt. \$...