Banach Fixed Point and Differential Equations

In summary, the problem requires finding the value of x, correct to three decimal places, for which the given integral is equal to 1/2. The solution can be obtained using Banach's fixed point theorem or Picard's method, but it is also possible to solve the problem by directly integrating the given function. Using the substitution t=tanθ, the integral can be simplified and solved for x, resulting in a solution of x=1.4743.
  • #1
ChemEng1
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Homework Statement


Find the value of x, correct to three decimal places for which: [itex]\int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\frac{1}{2}[/itex].

Homework Equations


Banach's Fixed Point Theorem
Picard's Theorem?

The Attempt at a Solution


I'm not sure where to start with this type of problem.

From other BFPT problems, I will need to show a contraction mapping into itself.

Any pointers would be greatly appreciated.
 
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  • #2
Banach's fixed point theorem and Picard's method show that this problem has a unique solution. Actually finding the solution doesn't require that.

You could, for example, actually integrate that:
[tex]\int_0^x \frac{t^2}{1+ t^2}dt= \int_0^x 1- \frac{1}{t^2+ 1}dt[/itex]
is pretty easy to integrate.
 
  • #3
[itex]\int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\int^{x}_{0}(1-\frac{1}{1+t^{2}})dt[/itex]

Let t=tanθ. Then dt=sec2θdθ.

[itex]\int^{x}_{0}(1-\frac{1}{1+t^{2}})dt=\int^{x}_{0}dt-\int^{x}_{0}\frac{1}{sec^{2}\vartheta}[/itex]sec2θdθ=[itex]\int^{x}_{0}dt-\int^{x}_{0}[/itex]dθ=t-arctan(t) from 0 to x=x-arctan(x).

x-arctan(x)=.5 at x=1.4743.

Are you familiar with any approaches that use BFPT?
 
Last edited:

Related to Banach Fixed Point and Differential Equations

1. What is the Banach Fixed Point Theorem?

The Banach Fixed Point Theorem states that for a complete metric space, any contraction mapping has a unique fixed point. This means that there exists a point in the space that does not move under the mapping.

2. How is the Banach Fixed Point Theorem used in differential equations?

The Banach Fixed Point Theorem is used in differential equations to prove the existence and uniqueness of solutions. It is particularly useful in proving the existence of solutions for nonlinear differential equations.

3. What is a contraction mapping?

A contraction mapping is a function from a metric space to itself that decreases the distance between points. This means that for any two points in the space, the distance between them after applying the function is always smaller than the distance between them before.

4. Can the Banach Fixed Point Theorem be applied to all differential equations?

No, the Banach Fixed Point Theorem can only be applied to certain types of differential equations, specifically those that can be written in the form of a contraction mapping. It cannot be applied to all differential equations.

5. Are there any limitations to the Banach Fixed Point Theorem?

Yes, the Banach Fixed Point Theorem has several limitations. It can only be applied to complete metric spaces, and the contraction mapping must be continuous. Additionally, the theorem only guarantees the existence of a fixed point, not its actual value.

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