When dealing with any mathematical problem, often the best way to do it is to break it up into smaller parts that are easier to deal with. So starting with our original linear inhomogeneous IVP,
We consider the simpler problems
(And, in order for uniqueness, all problems are coupled with the additional physical constraint that the integral of
If we have found a solution
What is nice about this approach is that (1) and (2) are already well studied problems. For the first, we use the fundamental solution of the heat equation, also known as the heat kernel, and for the second, we use the Green's function (see item 14 in the linked table).
In the following, assume
The fundamental solution is the solution
This is a very well known problem, and the solution is
The Green's function is the solution
And again, the solution of this problem is also well known,
Now that these two solutions are known, we can write
In your case with
Though often this procedure will not give you closed forms all that easily, it is the best way to deal with these kinds of problems in general, because typically closed forms are just not possible. To get closed forms, one would usually use separation of variables, but I stress that, in general, techniques like separation of variables will just fail.
What is 4 over 16 simplified to?
Using impulse formula, derive p_1 i+p_2 i=p_1 f+p_2 f formula where i-is initial state, f-is the final state after collision. Thanks
Averaging Newton's Method and Halley's Method.
Are these two methods identical? / Do they return the same result per iteration?
Difference between Newton's method and Gauss-Newton method
Prove that Newton's Method applied to converges in one step? Would it be because the derivative of is simply ?
A tangent line through the origin has equation . If it meets the graph at , then and . Therefore, .
Use Newton's Method to solve for
Using Newton's method below:
using this chord formula where the chord length is cm:
supposing the radius is cm and the angle is unknown, show the iterative Newton's Method equation you would use to find an approximate value for in the context of this problem (using the appropriate function and derivative).
Estimate the number of iterations of Newton's method needed to find a root of to within .
Why the bisection method is slower than Newton's method from a complexity point of view?
To find approximate we can use Newton's method to approximately solve the equation for , starting from some rational .
Newton's method in general is only locally convergent, so we have to be careful with initialization.
Show that in this case, the method always converges to something if .
Consider the function . Consider applying Newton's Method for minimizing f. How many iterations are needed to reach the global minimum point?
1. If you weigh 140 lbs on Earth, what is your mass in kilograms
2.Using the answer from problem I (hopefully it's correct), determine your weight in Newton's if you were on the moon. Yes, you have to look up something to complete this problem.
3. You measured the mass of a rock to be 355g. What is its weight?
4.You are now holding the rock in your hand from problem 3. What force is the rock on your hand? How much force do you need to exert on the rock to hold in stationary.
A solution containing 5.24 mg/100 mL of A (335 g/mol) has a transmittance of 55.2% in a 1.50-cm cell at 425 nm.find molar absorbtivity
Find the cube root of 9, using the Newton's method.