# The numerical solution of ordinary and partial differential equations pdf

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## chapter and author info

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations ODEs. Their use is also known as " numerical integration ", although this term can also refer to the computation of integrals. Many differential equations cannot be solved using symbolic computation "analysis".

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Over the last hundred years, many techniques have been developed for the solution of ordinary differential equations and partial differential equations. While quite a major portion of the techniques is only useful for academic purposes, there are some which are important in the solution of real problems arising from science and engineering. In this chapter, only very limited techniques for solving ordinary differential and partial differential equations are discussed, as it is impossible to cover all the available techniques even in a book form.

The readers are then suggested to pursue further studies on this issue if necessary. After that, the readers are introduced to two major numerical methods commonly used by the engineers for the solution of real engineering problems. Dynamical Systems - Analytical and Computational Techniques. To begin with, a differential equation can be classified as an ordinary or partial differential equation which depends on whether only ordinary derivatives are involved or partial derivatives are involved.

The differential equation can also be classified as linear or nonlinear. In Eq. The general solution of non-homogeneous ordinary differential equation ODE or partial differential equation PDE equals to the sum of the fundamental solution of the corresponding homogenous equation i.

A nonlinear differential equation is generally more difficult to solve than linear equations. 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. 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.

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 linear differential equation is generally governed by an equation form as Eq. Truly nonlinear in the sense that F is nonlinear in the derivative terms.

Quasi-linear 1st PDE if nonlinearity in F only involves u but not its derivatives. Quasi-linear 2nd PDE if nonlinearity in F only involves u and its first derivative but not its second-order derivatives. Many engineering problems are governed by different types of partial differential equations, and some of the more important types are given below.

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.

For other numerical methods such as boundary element method, they are less commonly adopted by the engineers; hence, these methods will not be discussed in this chapter. For equations which can be expressed in separable form as shown below, the solution can be obtained easily as.

This quadratic equation in y 2 can be solved with two solutions by the quadratic equation as. Since the second solution does not satisfy the boundary condition, it will not be accepted; hence, the solution to this differential equation is obtained. For the following equation form, it is possible to solve it by variations of parameters. Comparing the terms, it gives. Substitute it to the ODE. The Bernoulli equation is an important equation type which can be solved in a similar way by variation of parameters.

Consider the following form of equation. Inverting z to get y. For equation of the following type, where all the coefficients are constant, it can be evaluated according to different conditions.

The resulting non-linear ODE is hence separable and can be solved implicitly. There are various tricks to solve the differential equations, like integration factors and other techniques. A very good coverage has been given by Polyanin and Nazaikinskii [ 29 ] and will not be repeated here. The purpose of this section is just for illustration that various tricks have been developed for the solution of simple differential equations in homogeneous medium, that is, the coefficients are constants inside a continuous solution domain.

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. 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. 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.

To begin with, let us consider a review of conic curves ellipse, parabola and hyperbola. Following the conic curves, the general partial differential is also classified according to similar criterion as. This classification was proposed by Du Bois-Reymond [ 41 ] in In this section, only some of the more common techniques are discussed, and the readers are suggested to read the works of Hillen et al.

For soil consolidation problem, the governing conditions are given by. Assuming variable u x, t can be separated, using separation of variables. The eigenvalues are given by. The fundamental solutions are then expressed as.

One-dimensional 1D wave equation appears in many physical and engineering problems. 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.

Consider u x, t is given by X x T t. The wave equation will give. Laplace equation forms an important governing condition for many types of problems. Some of the more common forms are given by. Neumann problem: normal derivative u x or u y are usually prescribed on the boundary for many mathematical problems. This case can be solved by the use of complex analysis or series method for which many analytical solutions are available in the literature.

In many anisotropic seepage problems, however, the normal of a derived quantity at any arbitrary direction seepage flow normal to an impermeable surface is 0 instead of u x or u y being zero.

For such cases, it is very difficult to obtain the analytical solution if the solution domain is nonhomogeneous, and the use of numerical method such as the finite element method appears to be indispensable. 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.

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.

Many engineering problems fall into such category by nature, and the use of numerical methods will be necessary. Currently, there are several major numerical methods commonly used by the engineers: finite difference method, finite element method, boundary element method and distinct element. There are also other less common numerical methods available for practical problems, and many researchers also try to combine two or even more fundamental numerical methods so as to achieve greater efficiency in the analysis.

In general, the solution domain is discretized into series of subdomains with many degrees of freedom. The number of variables or degrees of freedom may even exceed millions for large-scale problems, and sometimes very special material properties are encountered so that the system is highly sensitive to the method of discretization and the method of solution. Similar to the ODE and PDE, it is impossible to discuss the details of all the numerical methods and the author choose to discuss the finite element method due to the wide acceptance of the method and this method is more suitable for general complicated methods.

Except for some simple problems with regular geometry and loading, it is very difficult to solve most of the boundary value problems with the yield of analytical solutions. Towards this, the use of numerical method seems indispensable, and the finite element is one of the most popular methods used by the engineers [ 32 , 38 ]. There are two fundamental approaches to FEM, which are the weighted residual method WRM and variational principle, but there are also other less popular principles which may be more effective under certain special cases.

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. Specifically, in elasticity for instance, the principle of virtual work including both principle of virtual displacement and virtual stress is considered to be the weak form of the equivalent integration for the governing equilibrium equations.

Furthermore, the aforementioned weak form of equivalent integration on the basis of the Galerkin method can also be evolved to a variation of a functional if the differential equations have some specific properties such as linearity and selfadjointness. Principles of minimum potential energy and complementary energy are two variational approaches equivalent to the fundamental equations of elasticity.

Since displacement is usually the basic unknown quantity in FEM, only the principle of virtual displacement and minimum potential energy will be introduced in the following section.

There are other ways to form the basis of FEM with advantages in some cases, but these approaches are less general and will not be discussed here.

The principle of virtual displacement is the weak form of the equivalent integration for equilibrium equations and force boundary conditions. Given the equilibrium equations and force boundary conditions in index notation,. The weak form of Eq. It can be seen clearly from Eq. In other words, the summation of the internal and external virtual works is equal to 0, which is called the principle of virtual displacement. Under this case, we can conclude that a force system will satisfy the equilibrium equations if the summation of the work done by it under any virtual displacement and strain is equal to 0.

Based on Eq. Due to the symmetry of the constitutive matrix D i j k l , we can further obtain. Given the assumptions in linear elasticity. The solution of a general continuum problem by FEM always follows an orderly step-by-step process which is easy to be programmed and used by the engineers.

For illustration, a three-node triangular element for plane problems is taken as an example to illustrate the general expressions and implementation procedures of FEM. The first step in the finite element method is to divide the structure or solution region into subdivisions or elements.

Hence, the structure is to be modelled with suitable finite elements. In general, the number, type, size, and arrangement of the elements are critical towards good performance of the numerical analysis. A typical discretization with three-node triangular element is shown schematically in Figure 1. Mesh generation can be a difficult process for a general irregular domain. If only triangular element is to be generated, this is a relatively simple work, and many commercial programs can perform well in this respect.

There are also some public domain codes EasyMesh or Triangle written in C which are sufficient for normal purposes.

## Numerical Solution of Ordinary and Partial Differential Equations

In mathematics , a partial differential equation PDE is an equation which imposes relations between the various partial derivatives of a multivariable function. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research , in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations. Partial differential equations are ubiquitous in mathematically-oriented scientific fields, such as physics and engineering. For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion , electrostatics , electrodynamics , fluid dynamics , elasticity , general relativity , and quantum mechanics. Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise.

### chapter and author info

Over the last hundred years, many techniques have been developed for the solution of ordinary differential equations and partial differential equations. While quite a major portion of the techniques is only useful for academic purposes, there are some which are important in the solution of real problems arising from science and engineering. In this chapter, only very limited techniques for solving ordinary differential and partial differential equations are discussed, as it is impossible to cover all the available techniques even in a book form. The readers are then suggested to pursue further studies on this issue if necessary. After that, the readers are introduced to two major numerical methods commonly used by the engineers for the solution of real engineering problems.

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