# e-book Partial Differential Equations

We will discuss the qualitative behaviour of solutions and, thus, be able to classify the most important partial differential equations into elliptic, parabolic, and hyperbolic type. Possible initial and boundary conditions and their impact on the solutions will be investigated. Solution techniques comprise the method of characteristics, Green's functions, and Fourier series.

Aims : To introduce the basic phenomenology of partial differential equations and their solutions. To construct solutions using classical methods.

### References

Objectives : At the end, you will be familiar with the notion of well-posed PDE problems and have an idea what kind of initial or boundary conditions may be imposed for this purpose. You will have studied some techniques which enable you to solve some simple PDE problems.

You will also understand that properties of solutions to PDEs sensitively depend on the its type. Books : A script based on the lecturer's notes will be provided. For further reading you may find the following books useful sections of relevance will be pointed out in the script or in the lectures :. S Salsa: Partial differential equations in action, from modelling to theory. Nevertheless, some techniques can be used for several types of equations.

The h -principle is the most powerful method to solve underdetermined equations. The Riquier—Janet theory is an effective method for obtaining information about many analytic overdetermined systems. The method of characteristics can be used in some very special cases to solve partial differential equations. In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods.

Many interesting problems in science and engineering are solved in this way using computers , sometimes high performance supercomputers. From Sophus Lie 's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups , be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration.

He also emphasized the subject of transformations of contact. A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions Lie theory. Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines. The Adomian decomposition method , the Lyapunov artificial small parameter method, and He's homotopy perturbation method are all special cases of the more general homotopy analysis method.

These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known perturbation theory , thus giving these methods greater flexibility and solution generality. The three most widely used numerical methods to solve PDEs are the finite element method FEM , finite volume methods FVM and finite difference methods FDM , as well other kind of methods called Meshfree methods , which were made to solve problems where the before mentioned methods are limited.

The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. The finite element method FEM its practical application often known as finite element analysis FEA is a numerical technique for finding approximate solutions of partial differential equations PDE as well as of integral equations. The solution approach is based either on eliminating the differential equation completely steady state problems , or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge—Kutta, etc.

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Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives. Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry.

## MA3G1 Theory of Partial Differential Equations

In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design. From Wikipedia, the free encyclopedia.

Main article: First-order partial differential equation. Main article: Separable partial differential equation. Main article: Method of characteristics. Main article: Fundamental solution. Further information: Superposition principle. See also: nonlinear partial differential equation. Main article: Finite element method. Main article: Finite difference method. Main article: Finite volume method.

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Modelling and Control of Robot Manipulators. The nature of mathematical modeling Reprinted with corr. Cambridge: Cambridge Univ. Adomian, G. Solving Frontier problems of Physics: The decomposition method. Kluwer Academic Publishers. Courant, R. Evans, L.

Elements of partial differential equations [Online-Ausg. Berlin: de Gruyter. John, F. Jost, J. Lewy, Hans , "An example of a smooth linear partial differential equation without solution", Annals of Mathematics , Second Series, 66 1 : —, doi : A fairly general method for constructing solutions to the Laplace equation 14 involves the following. Any semilinear partial differential equation of the second-order with two independent variables 10 can be reduced, by appropriate manipulations, to a simpler equation that has one of the three highest derivative combinations specified above in examples 11 , 12 , and It is apparent that the transformed equation 17 has only one highest-derivative term, just as the heat equation These integrals determine two different families of real characteristics.

Apart from notation, the left-hand side of the last equation coincides with that of the wave equation In this case the general integrals of equations 15 and 16 are complex conjugates; these determine two families of complex characteristics. Most PDEs of mathematical physics govern infinitely many qualitatively similar phenomena or processes. This follows from the fact that differential equations have, as a rule, infinitely many particular solutions. The specific solution that describes the physical phenomenon under study is separated from the set of particular solutions of the given differential equation by means of the initial and boundary conditions.

For simplicity and clarity of illustration, the basic problems of mathematical physics will be presented for the simplest linear equations 11 , 12 , and 14 only.

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In addition, all problems will be supplemented with some boundary conditions as given below. First boundary value problem. Second boundary value problem. Third boundary value problem. Mixed boundary value problems. Solutions to the above initial-boundary value problems for the heat equation can be obtained by separation of variables Fourier method in the form of infinite series or by the method of integral transforms using the Laplace transform.

For other linear heat equations, their exact solutions, and solutions to associated Cauchy problems and boundary value problems, see Linear heat equations at EqWorld. Boundary value problems. In addition, appropriate boundary conditions, 19 , 20 , 21 , or 22 , are imposed. Solutions to these boundary value problems for the wave equation can be obtained by separation of variables Fourier method in the form of infinite series.

Goursat problem. For other linear wave equations, their exact solutions, and solutions to associated Cauchy problems and boundary value problems, see Linear hyperbolic equations at EqWorld. Setting boundary conditions for the first, second, and third boundary value problems for the Laplace equation 14 means prescribing values of the unknown function, its first derivative, and a linear combination of the unknown function and its derivative, respectively. For elliptic equations, the first boundary value problem is often called the Dirichlet problem , and the second boundary value problem is called the Neumann problem.

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For other linear elliptic equations, their exact solutions, and solutions to associated boundary value problems, see Linear elliptic equations at EqWorld. Equation 28 is also called a heat equation with a nonlinear source. This equation is used for describing wave processes in gas dynamics, hydrodynamics, and acoustics. Equation 30 can be reduced to a linear equation see Polyanin and Zaitsev, Equations of this form arise in differential geometry and various areas of physics superconductivity, dislocations in crystals, waves in ferromagnetic materials, laser pulses in two-phase media, and others.

The following classes of solutions are usually regarded as exact solutions to nonlinear partial differential equations of mathematical physics:. The simplest types of exact solutions to nonlinear PDEs are traveling-wave solutions and self-similar solutions. They often occur in various applications. Consequently, a Cartesian coordinate system moving with a constant speed can be introduced in which the profile of the desired quantity is stationary.

The profiles of these solutions at different time instants are obtained from one another by a similarity transformation like scaling. The Cauchy problem and boundary value problems for nonlinear equations are stated in exactly the same way as for linear equations see Basic Problems for PDEs of Mathematical Physics.

The Cauchy problem for a nonlinear heat equation is stated as follows: find a solution to equation 27 subject to the initial condition The first boundary value problem for a nonlinear wave equation as follows: find a solution to equation 32 subject to the initial conditions 18 and the boundary conditions Apart from second-order PDEs, higher-order equations also quite often arise in applications.

Below are only a few important examples of such equations with some of their solutions. For solutions to associated Cauchy problems and boundary value problems, see Equation of transverse vibration of elastic rods at EqWorld. For solutions to associated boundary value problems, see Biharmonic equation at EqWorld. In particular, the mathematical modeling of moderate-amplitude shallow-water surface waves is based on this equation.