# Question:How to make 2nd derivative smoother?

## Question:How to make 2nd derivative smoother?

Maple 2018

The issue concerns calculation of 2nd derivative of the numerical solution of 2nd order BVP for an ordinal DE.

Lets consider a test problem:

-y’’(t)+y(t)=t*sin(5*t)

y(0)=0.2, y(1)=1

Numerical solution was obtained by

`dsolve([-diff(F(x), x\$2)+F(x)= x*sin(5*x), F(0)=0.2, F(1)=1], [F(x)], type = numeric, 'output' = Array([seq(k/5, k=0..5)]));`

dsolve returns the values of the solution and its 1st derivative (for the test example, a solution can be obtained analytically, but for general case I require numerical solution)..

It is needed to calculate the 2nd derivative of the solution.

I tried to use the Bessel method, when the solution’s 2nd derivative is calculated as the 2nd derivative of 5th degree polynomial having in consequent points t1, t2, t3 the values y(t1), y(t2), y(t3) and 1st derivatives y’(t1), y’(t2), y’(t3), all obtained from the numerical solution. Later I use Hermit piecewise-polynomial interpolation, based on values of the solution and its 1st and 2nd derivatives (polynomial of 5th degree).

Unfortunately, 2nd derivative of such interpolation has a large non-smoothness.

Here, we see the solution. It is smooth.

Here, 1st derivative. Still smooth enough.

Here, 2nd derivative. Typical view with large and sharp teeth.

Maybe, there exists a simple method for calculation of more smooth 2nd derivative?

Also. it is desirable that it would be embedded in Maple.

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