Real analysis is a mathematical branch that studies the properties of real numbers, rows, series, limit, continuity, derivative, integral. Real analysis provides a strong foundation to understand the basic concepts of calculus and to be the basis for many other mathematical branches.

Integral equation:

Integral equation is a mathematical equation in which the unknown is a function, and that function appears under the integral sign. Integral equations often appear in different areas like physics, engineering, economics and statistics.

Real Analysis Connection with Integral Equal

Real analysis provides very important tools for understanding and solving integral equations. Some key concepts in real analysis are very relevant to integral equations among others:

• Integral Riemann and Lebesgue: This integral concept is the basis of integral equations. Integral is used to calculate the area below the curve, which is a fundamental concept in many integral equations.
• Function Ranges: Many integral equations solutions can be expressed in a series of functions. The concept of series convergence is crucial in determining whether the solution is valid.
• Function Room: Integral equations often lie within certain function spaces (for example, continuu function space, Hilbert space). The concept of norm and distance in function space is very useful in analyzing the properties of solutions.
• Fixed Point Theorem: The fixed point theorem, like the Banach theorem, is used to prove its existence and uniqueness of integral equations.

Integral Equal Solving Method

There are some methods that can be used to solve integral equations, among other things:

• Institute Method: Turning integral equations into easier differential equations to solve.
• Laplace Transformation Method: Transform integral equations into an algebra equation in the Laplace domain, then solve them and transform Laplace invers.
• Drag Method: Looking for solutions in rank series or Fourier series.
• Numeric Method: Using numeric methods such as the quadramic method to calculate numeric value of integral.

Integral Equal Applications

Integral equations have a lot of applications in different fields, like:

• Physics: Difusion equations, wave equations, potential equations.
• Engineering: The voltage analysis in the structure, the fluid calculation.
• Economics: The model of economic growth, the portfolio theory.
• Statistics: Estimated parameters in statistical models.

Simple Example

The Volterra equation is one of the examples of integral equations often encountered:

y(x) = f(x) + ∫[a,x] K(x,t)y(t)dt

where:

• y (x) is an unknown function
• f (x) is a known function
• K (x, t) is integral kernel

Conclusion

Real analysis provides a strong foundation for understanding and solving integral equations. Conceptions like integral, series, and space functions are very important in integral equation analysis. The integral equation itself has many important applications in different areas of science and engineering.

Source: Stakgold, I. Green's Functions and Boundary Value Programs. Wiley.