Differential Equations Solved Problems

In general, the solution domain is discretized into series of subdomains with many degrees of freedom.

In general, the solution domain is discretized into series of subdomains with many degrees of freedom.

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It is common that nonlinear equation is approximated as linear equation (over acceptable solution domain) for many practical problems, either in an analytical or numerical form.

The nonlinear nature of the problem is then approximated as series of linear differential equation by simple increment or with correction/deviation from the nonlinear behaviour.

Differential equation can further be classified by the order of differential.

In general, higher-order differential equations are difficult to solve, and analytical solutions are not available for many higher differential equations.

For example, a vibrating string or pile driving process is given by this type of differential equation.

This problem is also commonly solved by the method of separation of variables In general, analytical solutions are not available for most of the practical differential equations, as regular solution domain and homogeneous conditions may not be present for practical problems.

The readers are also suggested to read the works of Greenberg [14], Soare et al. There are many elegant tricks that have been developed for the solution of different forms of differential equations, but only very few techniques are actually used for the solution of real life problems.

In many engineering or science problems, such as heat transfer, elasticity, quantum mechanics, water flow and others, the problems are governed by partial differential equations.

A linear differential equation is generally governed by an equation form as Eq. There are many methods of solutions for different types of differential equations, but most of these methods are not commonly used for practical problems.

In this chapter, the most important and basic methods for solving ordinary and partial differential equations will be discussed, which will then be followed by numerical methods such as finite difference and finite element methods (FEMs).