Engineering Applied Exercise Finite Difference Method|My homework helper

Posted: January 27th, 2023

*You must show sufficient detail to support your work to  earn credit for your calculations. This can be hand-written work, typed  calculations, or excel formulas. To avoid round-off error, retain  at least six decimal places in all of your calculations.  Complete  all trigonometric calculations in

Knowing the value of a two-dimensional function at each node in the mesh, your objective is to calculate the partial derivatives and at node

[Note that, in this example, the mesh sizes in x and y are identical (h); strictly speaking, this need not be true. In some applications, we may need more resolution along the x- or y-axis; we could then use separate mesh sizes hx and hy.]

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By definition, the partial derivative of a function with respect to x

and the partial derivative with respect to y is

If we applied these formula to our grid values, we would get the finite difference expressions

Note that these are approximations to the values of the partial derivatives since we’re not taking the limit as h goes to zero; but as h becomes smaller, the approximations should improve.

With this background, here’s your assignment:

*You must show sufficient detail to support your work to earn credit for your calculations. This can be hand-written work, typed calculations, or excel formulas. To avoid round-off error, retain at least six decimal places in all of your calculations.  Complete all trigonometric calculations in radians.

· Assume the function f is defined as f(x, y) = 3 tan x cos y

· Use differentiation rules to find the exact partial derivatives and , and evaluate those exact partial derivatives at (1.56, -2.1).

· Use the finite difference formulas to estimate and , at (1.56, -2.1) for three different values of the mesh size

· h = 0.01

· h = 0.001

· h = 0.0001

· Use your calculated values to fill in this table:

· Answer the following questions:

· For which partial derivative ( or is the finite difference approximation consistently more accurate? Why do you think the finite difference approximation for the other partial derivative is consistently less accurate?  (Hint: What happens to the tan x function near x = 1.56?). Under what real-world conditions might we see such poor approximations?

SOLUTION

The finite difference method is a numerical method used to solve differential equations by approximating the derivative of a function at discrete points. It involves dividing the domain of the function into a finite number of points, and then approximating the derivative at each point using the values of the function at the surrounding points.

This method is commonly used in engineering and physics to solve problems involving partial differential equations and boundary value problems. It is relatively easy to implement and can be used to approximate solutions to problems that have no exact analytical solution.