# Fourth order boundary value problem

• AlexCdeP
In summary, the algorithm is as follows: 1. Solve for y(1) and y(5) using y(1)=y''(1) = 0. 2. Record values of y''(5) and y'''(5). 3. Solve for y(1) and y(5) using y(1)=1 and y(1)=0. 4. Record values of y''(5) and y'''(5).
AlexCdeP
Hi guys, so I'm stuck on quite an interesting problem, and have been for a few days now. If anybody can take the time to have a look at it that would be the most incredible thing ever, because I have reached a point where I am at a loss.

Solve the following 4th order differential equation

20x^4y''''(x) - x^4y'''(x) + 3x^2y''(x) - 6xy'(x)+ 6y(x) =0

subject to the boundary conditions: y'(1)=0, y'''(1) = 1, y''(5) = -50,
and y'''(5)= -20. The superscripts within parentheses indicate the order
of the derivatives. Provide a plot of your solution y(x) from x = 1 to
x = 5.

I won't post my MATLAB code right now because it's very long, and I doubt any of you will want to wade through it! If anybody wants me to post it just say so. Instead I will write down what I think the algorithm is and I want you to call me out if you think I've made a mistake.

1. Rearrange the ode to solve for y'''' and then convert each differential into first order ODE. We have
function F = dEqs(y,x) % First-order differential
F=zeros(4,1);
F(1)=y(2);
F(2)=y(3);
F(3)=y(4);
F(4)= (1/20)*y(4)-(3/(20*x^2))*y(3)+(3/(10*x^3))*y(2)-(3/(10*x^4))*y(1);

2. Now this can at least be solved by ode45, BUT I don't have initial conditions, so I must guess my initial conditions. I need y''(1) and y(1), so I guess these (how I make a good guess I don't really know). This is called the shooting method for those who are interested. I can't use bvp4c because I need 3 out of 4 initial conditions I think. If I had 3 out of 4 I could solve it but...

3. THIS IS WHERE I AM STUCK I start with guessing y''(1) but I can't check this corresponds to y'(5) when I integrate because I don't know it. I can't guess y(1) for the same reason, in fact I don't know how to deal with that at all.

Any idea how I should go about solving this problem? I have a code for solving a second order ode, third or fourth with more forgiving boundary conditions... anybody?

Thank you!

First of all, is this homework? If it is, you have posted in the wrong forum.

AlexCdeP said:
I start with guessing y''(1), but I can't check that this corresponds to y'(5) ... because I don't know it.
I think you do not need to know y'(5). Start with guessing y(1), and let it correspond to y''(5). Also, guess y''(1), and let it correspond to y'''(5). Then see if your y''(1) guess corresponds to y'''(5). If not, adjust your y''(1) guess. Continue this process until you hit the y'''(5) target, then see if your y(1) guess corresponds to y''(5). If not, adjust your y(1) guess, and start all over.

(In the above, I am not sure if I should say iterate on your y''(1) guess last, or your y(1) guess last. Does anyone reading this thread have advice on the preferred sequence of events?)

Also, I tried to write a search algorithm in ISO C 1999 programming language, to search for two initial guesses, as you are doing. Where I currently put the project on hold (if I recall correctly) is, how to detect a "NaN" (not a number) value within my program, so I could change the search process accordingly. But when I perused the web, if I recall, it was still unclear to me how to detect NaN using portable, standard ISO C 1999 code. Does anyone have experience with how to do this using portable, standard C 1999, or advice?

nvn said:
I think you do not need to know y'(5). Start with guessing y(1), and let it correspond to y''(5). Also, guess y''(1), and let it correspond to y'''(5). Then see if your y''(1) guess corresponds to y'''(5). If not, adjust your y''(1) guess. Continue this process until you hit the y'''(5) target, then see if your y(1) guess corresponds to y''(5). If not, adjust your y(1) guess, and start all over.

(In the above, I am not sure if I should say iterate on your y''(1) guess last, or your y(1) guess last. Does anyone reading this thread have advice on the preferred sequence of events?)
Yes. I have some advice. It's a linear problem, so the boundary conditions at x =5 are going to be linear functions of the unknown boundary conditions at x =1. So start out by solving the problem with y(1)=y''(1) = 0. Record the values of y''(5) and y'''(5). Then re-solve the equations with y(1)=1 and y''(1)=0. Record the values of y''(5) and y'''(5). This will give you enough information to determine the partial derivatives of the boundary values at y = 5 with respect to y(1). Then re-solve the equations one more time, with y(1)= 0, y''(1)=1. Record the values of y''(5) and y'''(5). This will give you enough information to determine the partial derivatives of the boundary values at y = 5 with respect to y''(1). You can now write two linear algebraic equations in two unknowns to determine the unknown initial conditions required to make good on the boundary conditions at x = 5.

Chet

## 1. What is a fourth order boundary value problem?

A fourth order boundary value problem is a type of mathematical problem that involves finding a function or set of functions that satisfies a differential equation with four derivatives and boundary conditions at two or more points. It is commonly used in physics and engineering to model phenomena such as heat transfer, vibrations, and fluid flow.

## 2. How is a fourth order boundary value problem different from other types of boundary value problems?

A fourth order boundary value problem is different from other types of boundary value problems, such as first or second order problems, because it involves more derivatives. This means that the solution is more complex and requires more information, such as boundary conditions at multiple points, to fully determine the solution.

## 3. What are some real-world applications of fourth order boundary value problems?

Fourth order boundary value problems are used in a variety of real-world applications, such as modeling heat flow in materials, analyzing the behavior of vibrating structures, and studying the flow of fluids. They are also commonly used in engineering design and analysis, as well as in physics research and experimentation.

## 4. How are fourth order boundary value problems typically solved?

Fourth order boundary value problems can be solved using a variety of techniques, including analytical methods, numerical methods, and computer simulations. Analytical methods involve finding an exact solution using mathematical equations, while numerical methods use approximations and algorithms to find an approximate solution. Computer simulations use advanced software to model and solve the problem.

## 5. What are some challenges associated with solving fourth order boundary value problems?

One of the main challenges with solving fourth order boundary value problems is the complexity of the problem, which can make it difficult to find an exact solution. This often requires the use of advanced mathematical techniques and computational tools. Additionally, obtaining accurate boundary conditions and understanding the physical context of the problem can also be challenging.

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