Differential Equations Solved Problems

Differential Equations Solved Problems-64
By nature, this type of problem is much more complicated than the previous ordinary differential equations.There are several major methods for the solution of PDE, including separation of variables, method of characteristic, integral transform, superposition principle, change of variables, Lie group method, semianalytical methods as well as various numerical methods.Moreover, the solution domain may be indeterminate (free surface seepage flow), the displacement is large so that the solution may deform under motion, or in an extreme case part of the material may tear off from the main body with continuous formation and removal of contacts.

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Although the existence and uniqueness of solutions for ordinary differential equation is well established with the Picard-Lindelöf theorem, but that is not the case for many partial differential equations.

In fact, analytical solutions are not available for many partial differential equations, which is a well-known fact, particularly when the solution domain is nonregular or homogeneous, or the material properties change with the solution steps.

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.

A differential equation is termed as linear if it exclusively involves linear terms (that is, terms to the power 1) of ″, or higher order term.

A nonlinear differential equation is generally more difficult to solve than linear 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.In finite element analysis of an elastic problem, solution is obtained from the weak form of the equivalent integration for the differential equations by WRM as an approximation.Alternatively, different approximate approaches (e.g. collocation method, least square method and Galerkin method) for solving differential equations can be obtained by choosing different weights based on the WRM and the Galerkin method appears to be the most popular approach in general.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.This approach is adopted for the solution of many non-linear engineering problems.Without such procedure, most of the non-linear differential equations cannot be solved.[16], Salsa [33], Polyanin and Zaitsev [31], Bertanz [2], Haberman [15] and many other published texts.One-dimensional (1D) wave equation appears in many physical and engineering problems.


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