Integral equation is one of the most important branches in mathematics that appear in different disciplines of science, including physics, engineering, economics and biology. This equation often appears as part of the modeling phenomenon where information is spread in time or space, as in the theory of electromagnetic field, quantum mechanics, and population theory.

An integral equations is the equation in which an unknown function appears under the integral sign. This equation has a common form:

f (x) = g (x) + HezabK (x, s) dsf (x) = g (x) + lamggggint

Where:

• f (x) f (x) f (x) is an unknown function,
• g (x) g (x) g (x) is a known function,
• K (x, s) K (x, s) K (x, s) is a kernel function that connects the variable xx and sss,
• The lambda is a parameter, and
• Integral aaa and bbb boundaries can be fixed borders or variables.

Integral Equal Classification

Integral equation is classified based on some criteria:

1. Volterra and Fretacus equation:
   • Volterra equation: If the integral limit depends on xxx, for example from aaa to xxx, then the equation is called Volterra. It often comes about in time evolution, where early conditions affect future results.
   • Freelandic equationIf the integral boundary is fixed, then the equation is called the nefarious equation. This equation often appears in static problems.
2. Linier and Non- Linier equation:
   • Linear: Integral equation is called linear if function is f (x) f (x) f (x) (x) appears in linear form inside integral.
   • Nonlinear: If f (x) f (x) f (x) appears in nonlinear form, then it's classified as nonlinear.
3. Integral Homogen and Non@-@ Homogen equation:
   • HomogeneousIf the g (x) g (x) g (x) g (x) is 0, it's called homogeneous.
   • Non- homogeneous: If g (x) g (x) g (x) is not zero, then it's called nonhomogen.

Integral Equal Completion Method

There are several methods of solving integral equations, depending on the shape and nature of the equation facing:

1. Sucusesive Iteration Methods (Successive Projected): This method was used to solve the integral equations of Volterra and Fretacus. This approach involves the use of a closer initial solution, then the iteration to get a more accurate solution. This process is often called Picard method and useful for integral linear and nonlinear equations.
2. Laplace Transformation Method: Volterra integral equations often solved using Laplace transformation, that turns integral equations into algebra equations. Solutions of the algebra equation then converted back to domain time through Laplace transformation inversion. This method is very effective for the equations involving the boundaries to change.
3. Separable Kernel Method: If kernel function K (x, s) K (x, s) K (x, s) can be separated into two function products, which is K (x, s) = xx (x) K (x) s) = phi (x) psi (x, s): This approach is very helpful in solving Fremanic equations.
4. Freshman Method for the Freshman Equal: Fremanic equation has properties that allow the use of the series to solve it. By utilizing the K (x, s) K (x, s) K (x, s) kernel, solutions are often obtained in the form of an infinite series.
5. Fourier Transformation MethodThe Fourier transformation can be used to simplify integral equations by turning them into easier shapes to break. This technique is very effective when the K (x, s) K (x, s) K (x, s) depends on x (x - sx) difference, which often occurs in physics like heat diffusion.
6. Numeric MethodIn many cases, the analytical solution to integral equations is difficult or impossible to get. To that end, a numerical method like Gauss square method, Collocation method, or numeric iteration method used to approach the solution. These techniques are very useful in solving complex problems where kernel and function g (x) g (x) g (x) g (x) has no simple form.

Integral Equal Applications

Integral equations have various applications in science and engineering:

1. PhysicsIn field theory, integral equations are used to model intermediate particles in remote interactions. The classic example is Poisson integral equation which is used in electrostatic and gravity.
2. Difusion TheoryIn thermal physics, integral equations model heat spread in medium. For example, integral Fourier equation Used to study heat transfer.
3. Biology and Epidemiology.: integral equations are also used in model population growth and spread disease, where late effects or long-term effects need to be calculated.
4. Optical: In optics and wave theory, integral equations are used to model electromagnetic behavior in certain media, including the reflective and refraction process.

Conclusion

Integral equations are powerful tools in mathematical modeling of various phenomena in nature and technology. With various methods of completion, both analytic and numerical, integral equations offer a way to understand and solve problems that involve interaction between variables that are scattered in space or time. The vast application of this theory made it one of the most important fields of applied mathematics.

source: Polyanin, A. D., & Manzhirov, A. V. (2008). Handbook of Integral Equations. CRC Press.