The nonlinear system of equations is a set of equations involving not only a linear form, but also in the form of rank, logarithm, trigonometry, or other nonlinear functions. Non-linear equations are found in many different disciplines, such as physics, engineering, biology and economics, because they can model complex and dynamic phenomena. In this article, we're going to talk about some general method of solving nonlinear equations, and its applications in various areas.

Nonlinear Equal System characteristic

Unlike linear equations that have more direct and stable solutions, nonlinear equations often have more than one solution, It makes the process of solving nonlinear equations more complicated and requires more advanced methods. The example of a simple nonlinear equation is:

x2 + y2 = 1x ^ 2 + y ^ 2 = 1 × 2

Here, the solution cannot be obtained only by method of substitution or simple linear elimination. Some special techniques are required to overcome this complexity.

Nonlinear System Completion Method

Here are some of the methods that are often used to solve nonlinear system equations:

1. Newton-Raphson Method

The Newton-Raphson method is one of the most often used literate methods to solve nonlinear equations. It utilizes the local linear approach of nonlinear function by finding its cut point to the x-axis.

For nonlinear equations system with more than one variable, this method requires the use of Jacobian, which is the partial derivative matrix of the equation system. The Iterbf of the Newton-Raphson method works with the following formula: xk + 1 = xk [J (xk)]

Where J (xk) J (mathbf {x} _ k) J (xk} Hay) is the Jacobian matrix of the system at xkmathbf {x} _ kxk and F (xk) F (mathbf {x} _ k) F (xk Hay) is nonlinear function at that point.

This method converges very quickly if the initial solution is quite close to the actual solution, but it may not work if the initial solution is far from the desired solution or if Jacobian is not invertible.

2. Broyden Method

Broyden's method is a variant of the Newton-Raphson method that is more computing because it doesn't require a Jacobian recounting on each iteration. As a control method - Newton, this method estimates change in Jacobian based on previous iterations. Broyden's methods are very useful to large systems that require a lot of iterations, thus saving time and computing resources.

3. Bisection Method

Bisection methods are numerical methods that are simpler but slower. It's used to find the root of nonlinear equations by cutting the solution range iterally until it reaches the desired level of precision. Although simple, bixy methods are only suitable for nonlinear equations with one variable, or a case in which other variables can be eliminated.

4. Homotopi Method

The Homotopian method is a further technique that uses a continuous transformation approach between nonlinear equations systems and a simpler system. In this process, the solution of a simple system is slowly "converted" to a solution of a more complex system through a continuous path.

Nonlinear Common System Applications

The nonlinear system of equations has many practical applications in various disciplines. Some examples are:

1. Physics: In physics, many phenomena are modeled using nonlinear equations, such as fluid motion, theory of relativity, and quantum dynamics. For example, the Schrödinger equation in quantum mechanics is an example of a nonlinear system often used.
2. BiologyIn biology, nonlinear equations are used to model population interactions in the ecosystem, the spread of disease in the population, and the growth of cells in the network.
3. Economic: Economics often use nonlinear equations to describe the interaction between economic variables such as supply and demand, inflation, economic growth, and the influence of monetary policy.
4. EngineeringIn engineering, nonlinear equations are used in structure design, analysis of control systems, dynamic modeling, and stability analysis.

Conclusion

The nonlinear system of equations is a very important tool in modeling complex phenomena in different areas of science. Although the solution is more complicated than linear equations, methods like Newton-Raphson, Broyden, and homotopies allow efficient and accurate solutions. The mastery of this method is very important to anyone who works in a field involving applied mathematics and numerical modeling.

source: Burden, R. L., & Faires, J. D. (2010). Numerical Analysis. Brooks / Cole.