I know [itex]y[/itex] is a function of [itex]x[/itex] [i.e. [itex]y=f\left(x\right)[/itex]](adsbygoogle = window.adsbygoogle || []).push({});

with two known boundary conditions, that is [itex]f\left(x=A\right)=C[/itex]

and [itex]f\left(x=B\right)=D[/itex] where [itex]C[/itex] and [itex]D[/itex] are known constants

(please see figure). I do not know the form of this function and therefore

I am trying to find the form of [itex]y[/itex] as a function of [itex]x[/itex] over the

whole interval [itex]A<x<B[/itex]. I have an additional condition that is if

I discretize the interval I can obtain the folowing relation

[itex]g(A,c)=g(c,d)=g(d,e)=g(e,B)=E[/itex]

where [itex]g[/itex] is a known function of the given arguments and [itex]E[/itex] is

a known constant. I think this problem can be solved by using the

variational principle possibly with the use of Lagrange multipliers.

I did some initial attempts but I am not sure about the results. Can

you suggest a method (variational or not) that can solve this problem

so that we can obtain the form of [itex]y[/itex] as a function of [itex]x[/itex] over

the whole interval.

Many thanks in advance!

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# Form of function over interval

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