Analytical methods and numerical methods are two different ways used in various fields of study to solve problems and help analyze complex systems or situations. The fields where these methods are extensively used are mathematics, physics, engineering, and computer science. Both these methods solve the problems with different underlying principles or techniques.
Anything that happens in real life is a part of nature’s behavior. To understand nature’s behavior or phenomena, one has to rely on mathematics. The role of mathematics as a tool is to decode the underlying physical phenomena. For example, falling off an object from a height is an example of nature’s behavior as the earth pulls everything towards its center. To understand it, one has to use the equation of motions derived by physicists.
One needs to use differential equations or other mathematics methods to solve problems in engineering and science. If an equation mimics nature’s behavior, based on prevailing conditions, one has to solve that equation. Solution techniques are of two types namely analytical method and numerical method.
Table of Contents
Analytical method:
A direct method involves the use of closed-form (single unique solution) equations to get “exact solutions”. For example, finding the root of a quadratic equation. Another example of the analytical method is a solution to the fin problem in heat conduction or the solution of Laplace’s 2D heat conduction.
Analytical methods, also known as exact or closed-form solutions, involve obtaining mathematical expressions or formulas that directly solve a given problem. These analytical methods depend on various mathematical techniques such as algebraic manipulation, calculus, and differential equations to get the exact solutions.
Analytical methods provide a deeper understanding of the problem by revealing the underlying mathematical relationships and properties. They often yield explicit formulas that allow for direct computation and provide insights into the behavior of the system being analyzed.
Advantages of analytical method
One of the important advantages of analytical methods is their ability to provide exact solutions. When an analytical solution exists, it is generally preferred because it gives precise and exact results and can be evaluated at any point in the problem domain. This makes analytical methods highly accurate and reliable. Furthermore, analytical solutions often provide a concise representation of the problem, allowing for further analysis and interpretation.
Limitations of the analytical method
However, analytical methods have limitations. They are only applicable to problems that can be expressed in mathematical terms and for which suitable mathematical techniques exist. Many real-world problems are complex or involve nonlinear equations that do not have closed-form solutions. In such cases, analytical methods may not be feasible or may require simplifying assumptions that limit their accuracy. Additionally, deriving analytical solutions can be time-consuming and challenging for complex problems, requiring advanced mathematical skills and expertise.
Numerical method:
On the other hand, numerical methods, also known as computational or approximate methods, rely on numerical techniques to obtain solutions. These methods involve discretizing the problem domain and approximating the solution through iterative computations. Numerical methods are particularly useful for solving complex problems that lack analytical solutions or require substantial computational resources to derive them.
When it is difficult to get closed-form equations and if it uses an algorithm to solve the given differential equation by successive “approximation”, the method of that solution is known as the numerical method. For example, solving the fluid dynamics problems involving the use of Navier Stoke’s equations by finite volume method or solving the structural problems by finite element method. One can use a computer to solve the linear equations obtained from the governing equations of the problem. Runge Kutta methods and Finite difference methods are also numerical methods.
Numerical methods work by dividing the problem domain into a discrete set of points or elements. This can also be termed meshing. The equations governing the problem are then approximated and solved iteratively using numerical algorithms for each cell of the mesh of the domain. These algorithms, such as finite difference, finite element, or finite volume methods, approximate the derivatives and integrals involved in the problem to obtain approximate solutions at discrete points. By iteratively refining these approximations, numerical methods converge toward the desired solution.
As the complexity of the science or engineering problem increases, the exact solution of mathematical expression is difficult to obtain. Hence, one has to rely on numerical methods like the finite difference method, finite volume method, finite element method, Runge Kutta method, Newton Raphson method, etc. Computers can do repetitive tasks very easily with the use of software like ANSYS-FLUENT, COMSOL, etc.
Advantages of the numerical method
The main advantage of numerical methods lies in their versatility and applicability to a wide range of problems. They can handle complex systems, nonlinear equations, and problems involving multiple dimensions. Numerical methods are most helpful in dealing with real-world problems like fluid flow, heat transfer, or electromagnetic fields. They allow researchers and engineers to simulate and analyze these phenomena by discretizing the problem domain and solving it computationally.
Limitations of the numerical method
However, numerical methods are not without limitations. They introduce errors due to the approximations made during the discretization process and the use of iterative computations. These errors can accumulate and affect the accuracy of the final solution. Furthermore, numerical methods require computational resources, including processing power and memory, especially for large-scale problems. The efficiency and convergence of numerical algorithms also depend on the properties of the problem being solved, such as its smoothness, linearity, or stiffness.
Difference between the analytical method and the numerical method
In summary, analytical methods and numerical methods offer distinct approaches to problem-solving and analysis. Analytical methods provide exact solutions when available, relying on mathematical techniques to derive closed-form expressions. They offer accuracy, insight, and concise representations of the problem. However, they are limited to problems with suitable mathematical formulations and can be challenging for complex scenarios.
Numerical methods, on the other hand, approximate solutions through iterative computations and discretization of the problem domain. They are versatile, applicable to a wide range of problems, and can handle complex systems and uncertainty. While they introduce errors and require computational resources, they provide practical solutions for problems lacking analytical solutions or requiring extensive computational analysis.
Let us take an example to understand the difference.
Quadratic Equation: ax^2 + bx + c = 0
-
Analytical Method: Analytical methods involve finding exact solutions using algebraic techniques. For a quadratic equation, you can use the quadratic formula to find the roots:
Quadratic Formula: x = (-b ± √(b^2 – 4ac)) / (2a)
Example: Let’s say we have the quadratic equation: 2x^2 – 5x + 2 = 0. Using the quadratic formula, you can calculate the roots analytically:
x1 = (-(-5) + √((-5)^2 – 4 * 2 * 2)) / (2 * 2) = (5 + √(25 – 16)) / 4 = (5 + √9) / 4 = (5 + 3) / 4 = 8 / 4 = 2
x2 = (-(-5) – √((-5)^2 – 4 * 2 * 2)) / (2 * 2) = (5 – √(25 – 16)) / 4 = (5 – √9) / 4 = (5 – 3) / 4 = 2 / 4 = 0.5
Using the analytical method, you’ve found the exact roots of the quadratic equation: x1 = 2 and x2 = 0.5.
-
Numerical Method: Numerical methods involve approximating solutions using numerical techniques, especially when you cannot find exact solutions analytically. One common numerical method for finding roots is the “Bisection Method.”
Example: Using the same quadratic equation: 2x^2 – 5x + 2 = 0, you can apply the bisection method as follows:
Start with an initial interval [a, b] such that the root lies between them (e.g., a = 0, b = 3).
- Calculate the midpoint c = (a + b) / 2.
- Evaluate the function f(c) at c.
- Determine in which subinterval [a, c] or [c, b] the root lies (based on the sign of f(c)).
- Update the interval [a, b] to the subinterval containing the root.
- Repeat steps 1-4 until you reach a desired level of accuracy.
Using the bisection method, you can iteratively approximate the roots. The final result would be an approximation, e.g., x ≈ 2 and x ≈ 0.5, similar to the analytical solutions.
In summary, the analytical method provides exact solutions using algebraic expressions, while numerical methods use iterative techniques to approximate solutions, especially when exact solutions are difficult or impossible to obtain. Numerical methods are valuable for solving a wide range of problems, including those with complex equations or functions where analytical solutions may not be readily available.
Thus, the selection between these two methods depends on the nature of the problem, the availability of analytical techniques, the required accuracy, and the computational resources at hand.
FAQs
What fundamental difference exists in problem-solving between the analytical method and the numerical method?
The analytical method uses mathematical equations to find precise, closed-form solutions, whereas the numerical method uses computations and algorithms to approximate solutions.
In comparison to numerical methods, how well do analytical methods handle complicated equations and functions?
In response, analytical methods excel at resolving precise solutions for a variety of issues in well-defined mathematical expressions and equations. For difficult, nonlinear equations where analytical solutions are elusive, numerical methods are more appropriate.
Can you explain the role of precision and approximation in these methods?
Analytical methods provide precise, symbolic solutions, whereas numerical methods involve approximation, making them suitable for solving problems that lack exact analytical solutions.
Which method is generally faster for solving problems: analytical or numerical?
Analytical methods are often faster when exact solutions exist. Numerical methods may require significant computational time, especially for problems involving extensive calculations or simulations.
In what situations would you prefer to use an analytical method over a numerical one, and vice versa?
Analytical methods are preferable when exact solutions are readily available, as they provide insight into the underlying mathematics. Numerical methods are chosen when dealing with complex physical systems, real-world data, or simulations where approximations are acceptable or necessary.