A question on integral equations

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In summary, the conversation discusses an integral equation with the operator K and a kernel, and the question of whether there is another operator R that can solve the equation when applied to the function f. The conversation also touches on the possibility of transforming integral equations into differential equations.
  • #1
eljose79
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Let be the integral equation:

f(x)-g(x)=Kf where K is the integral operator having the kernel

my question : is there an operator R satisfying:

g(x)-f(x)=Rg where f satisfy the original integral equation so it can be solved by mean of R operator.


Another question is can any integral equation be transformed into a differential equation?..if so how it is made?..thanks.
 
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  • #2
Originally posted by eljose79
Let be the integral equation:

f(x)-g(x)=Kf where K is the integral operator having the kernel

OK

my question : is there an operator R satisfying:

g(x)-f(x)=Rg where f satisfy the original integral equation so it can be solved by mean of R operator.

Just using algebra, I get:

1. f(x)-g(x)=Kf(x)
2. f(x)-Kf(x)=g(x)
3. (1-K)f(x)=g(x)
4. f(x)=(1-K)-1g(x)
5. g(x)-f(x)=[1-(1-K)-1]g(x)

Defining R as [1-(1-K)-1] seems to fit the bill. The only thing I am not sure of is whether (1-K) is invertible. I assumed that it was invertible in Step 4.

edit: fixed bracket
 
  • #3


To answer your first question, yes, there exists an operator R that satisfies g(x)-f(x)=Rg, where f satisfies the original integral equation. This operator is known as the inverse operator of K and is denoted as K^-1. It is defined as the operator that undoes the effect of K, i.e. K^-1Kf = f. So, if we apply K^-1 on both sides of the original integral equation, we get K^-1(f(x)-g(x))=K^-1(Kf), which simplifies to f(x)=K^-1(Kf). This shows that f satisfies the original integral equation and can be solved using the inverse operator K^-1.

To your second question, not all integral equations can be transformed into differential equations. However, there are certain integral equations that can be transformed into differential equations using a technique known as the Laplace transform. This transform converts an integral equation into a differential equation, which can then be solved using standard methods. The process of transforming an integral equation into a differential equation using the Laplace transform involves taking the Laplace transform of both sides of the integral equation and then simplifying the resulting equation to obtain a differential equation. This technique is commonly used in engineering and physics to solve integral equations.
 

1. What is an integral equation?

An integral equation is a mathematical equation that involves an unknown function in the form of an integral. It is a type of functional equation and is commonly used in physics, engineering, and other fields to model a variety of phenomena.

2. How is an integral equation different from a differential equation?

An integral equation involves the integration of an unknown function, while a differential equation involves the differentiation of an unknown function. In other words, an integral equation relates the value of a function at different points, while a differential equation relates the rate of change of a function at different points.

3. What are some applications of integral equations?

Integral equations are used in many fields to model various physical phenomena. Some common applications include heat transfer, fluid mechanics, electromagnetism, and quantum mechanics. They can also be used in image processing, signal processing, and finance.

4. How do you solve an integral equation?

The solution to an integral equation can be found by using various methods such as the method of successive approximations, the Fredholm method, or the Volterra method. The specific method used will depend on the type of integral equation and the properties of the function involved.

5. What are some challenges in solving integral equations?

Some challenges in solving integral equations include finding the appropriate method for a specific equation, determining the limits of integration, and dealing with singularities or other special cases. Additionally, integral equations can often be nonlinear, which makes them more difficult to solve compared to linear equations.

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