The Numerical Treatment of Differential Equations

This book constitutes an attempt to present in a connected fashion some of the most important numerical methods for the solution of ordinary and partial differential equations. The field to be covered is extremely wide, and it is clear that the present treatment cannot be remotely exhaustive; in particular, for partial differential equations it has only been possible to present the basic ideas, and many of the methods developed extensively by workers in applied fields - hydro- dynamics, aerodynamics, etc. -, most of which have been developed for specific problems, have had to be dismissed with little more than a reference to the literature. However, the aim of the book is not so much to reproduce these special methods, their corresponding computing schemes, etc. , as to acquaint a wide circle of engineers, physicists and mathematicians with the general methods, and to show with the aid of numerous worked examples that an idea of the quantitative behaviour of the solution of a differential equation problem can be obtained by numerical means with nothing like the trouble and labour that widespread prejudice would suggest. This prejudice may be partly due to the kind of mathe- matical instruction given in technical colleges and universities, in which, although the theory of differential equations is dealt with in detail, numerical methods are gone into only briefly.

I Mathematical preliminaries and some general principles.- Some notes on the numerical examples.- § 1. Introduction to problems involving differential equations.- 1.1. Initial-value and boundary-value problems in ordinary differential equations.- 1.2. Linear boundary-value problems.- 1.3. Problems in partial differential equations.- § 2. Finite differences and interpolation formulae.- 2.1. Difference operators and interpolation formulae.- 2.2. Some integration formulae which will be needed later.- 2.3. Repeated integration.- 2.4. Calculation of higher derivatives.- 2.5. Hermite’s generalization of Taylor’s formula.- § 3. Further useful formulae from analysis.- 3.1. Gauss’s and Green’s formulae for two independent variables.- 3.2. Corresponding formulae for more than two independent variables.- 3.3. Co-normals and boundary-value problems in elliptic differential equations.- 3.4. Green’s functions.- 3.5. Auxiliary formulae for the biharmonic operator.- § 4. Some error distribution principles.- 4.1. General approximation. “Boundary” and “interior” methods.- 4.2. Collocation, least-squares, orthogonality method, partition method, relaxation.- 4.3. The special case of linear boundary conditions.- 4.4. Combination of iteration and error distribution.- § 5. Some useful results from functional analysis.- 5.1. Some basic concepts of functional analysis with examples.- 5.2. The general theorem on iterative processes.- 5.3. The operator T applied to boundary-value problems.- 5.4. Problems of monotonic type.- 5.5. Application to systems of linear equations of monotonic type.- 5.6. Non-linear boundary-value problems.- II Initial-value problems in ordinary differential equations.- § 1. Introduction.- 1.1. The necessity for numerical methods.- 1.2. Accuracy in the numerical solution of initial-value problems.- 1.3. Some general observations on error estimation for initial-value problems.- I. Comparison of two approximations with different lengths of step.- II. The terminal check.- 1.4. Differential equations of the first order. Preliminaries.- 1.5. Some methods of integration.- 1.6. Error estimation.- I. Polygon method.- 1.7. Corresponding error estimates for the improved methods.- II. Improved polygon method.- III. Improved Euler-Cauchy method.- § 2. The Runge-Kutta method for differential equations of the n-th order.- 2.1. A general formulation.- 2.2. The special Runge-Kutta formulation.- 2.3. Derivation of the Runge-Kutta formulae.- 2.4. Hints for using the Runge-Kutta method.- 2.5. Terminal checks and iteration methods.- 2.6. Examples.- § 3. Finite-difference methods for differential equations of the first order.- 3.1. Introduction.- 3.2. Calculation of starting values.- 3.3. Formulae for the main calculation.- I. The Adams extrapolation method.- II. The Adams interpolation method.- III. Central-difference interpolation method.- IV. Mixed extrapolation and interpolation methods.- 3.4. Hints for the practical application of the finite-difference methods.- 3.5. Examples.- 3.6. Differential equations in the complex plane.- 3.7. Implicit differential equations of the first order.- § 4. Theory of the finite-difference methods.- 4.1. Convergence of the iterations in the main calculation.- 4.2. Convergence of the starting iteration.- 4.3. Recursive error estimates.- 4.4. Independent error estimates.- 4.5. Error estimates for the starting iteration.- 4.6. Systems of differential equations.- 4.7. Instability in finite-difference methods.- 4.8. Improvement of error estimates by use of a weaker Lipschitz condition.- 4.9. Error estimation by means of the general theorem on iteration.- § 5. Finite-difference methods for differential equations of higher order.- 5.1. Introduction.- 5.2. Calculation of starting values.- 5.3. Iterative calculation of starting values for the second-order equation y? = f (x,y,y?).- 5.4. Extrapolation methods.- 5.5. Interpolation methods.- 5.6. Convergence of the iteration in the main calculation.- 5.7. Principle of an error estimate for the main calculation.- 5.8. Instability of finite-difference methods.- 5.9. Reduction of initial-value problems to boundary-value problems.- 5.10. Miscellaneous exercises on Chapter II.- 5.11. Solutions.- III Boundary-value problems in ordinary differential equations.- § 1. The ordinary finite-difference method.- 1.1. Description of the finite-difference method.- 1.2. Examples of boundary-value problems of the second order.- I. A linear boundary-value problem of the second order.- II. A non-linear boundary-value problem of the second order.- III. An eigenvalue problem.- IV. Infinite interval.- 1.3. A linear boundary-value problem of the fourth order.- 1.4. Relaxation.- I. A linear boundary-value problem.- II. A non-linear boundary-value problem.- § 2. Refinements of the ordinary finite-difference method.- 2.1. Improvement by using finite expressions which involve more pivotal values.- 2.2. Derivation of finite expressions.- 2.3. The finite-difference method of a higher approximation.- 2.4. Basic formulae for Hermitian methods.- 2.5. The Hermitian method in the general case.- 2.6. Examples of the Hermitian method.- I. Inhomogeneous problem of the second order.- II. An eigenvalue problem.- 2.7. A Hermitian method for linear boundary-value problems.- § 3. Some theoretical aspects of the finite-difference methods.- 3.1. Solubility of the finite-difference equations and convergence of iterative solutions.- 3.2. A general principle for error estimation with the finite-difference methods in the case of linear boundary-value problems.- 3.3. An error estimate for a class of linear boundary-value problems of the second order.- 3.4. An error estimate for a non-linear boundary-value problem.- § 4. Some general methods.- 4.1. Examples of collocation.- 4.2. An example of the least-squares method.- 4.3. Reduction to initial-value problems.- 4.4. Perturbation methods.- 4.5. The iteration method, or the method of successive approximations.- 4.6. Error estimation by means of the general iteration theorem.- 4.7. Special case of a non-linear differential equation of the second order.- 4.8. Examples of the iteration method with error estimates.- I. A linear problem.- II. Non-linear oscillations.- 4.9. Monotonic boundary-value problems for second-order differential equations.- § 5. Ritz’s method for second-order boundary-value problems.- 5.1. Euler’s differential equation in the calculus of variations.- 5.2. Derivation of Euler’s conditions.- 5.3. The Ritz approximation.- 5.4. Examples of the application of Ritz’s method to boundary-value problems of the second order.- I. A linear inhomogeneous boundary-value problem.- II. An eigenvalue problem.- III. A non-linear boundary-value problem.- § 6. Ritz’s method for boundary-value problems of higher order.- 6.1. Derivation of higher order Euler equations.- 6.2. Linear boundary-value problems of the fourth order.- 6.3. Example.- 6.4. Comparison of Ritz’s method with the least-squares process.- § 7. Series solutions.- 7.1. Series solutions in general.- 7.2. Power series solutions.- 7.3. Examples.- § 8. Some special methods for eigenvalue problems.- 8.1. Some concepts and results from the theory of eigenvalue problems.- 8.2. The iteration method in the general case.- 8.3. The iteration method for a restricted class of problems.- 8.4. Practical application of the method.- 8.5. An example treated by the iteration method.- 8.6. The enclosure theorem.- 8.7. Three minimum principles.- 8.8. Application of Ritz’s method.- 8.9. Temple’s quotient.- 8.10. Some modifications to the iteration method.- 8.11. Miscellaneous exercises on Chapter III.- 8.12. Solutions.- IV Initial-and initial-/boundary-value problems in partial differential equations.- The need for a sound theoretical foundation.- § 1. The ordinary finite-difference method.- 1.1. Replacement of derivatives by difference quotients.- 1.2. An example of a parabolic differential equation with given boundary values.- 1.3. Error propagation.- 1.4. Error propagation and the treatment of boundary conditions.- 1.5. Hyperbolic differential equations.- 1.6. A numerical example.- 1.7. Graphical treatment of parabolic differential equations by the finite-difference method.- 1.8. The two-dimensional heat equation.- 1.9. An indication of further problems.- § 2. Refinements of the finite-difference method.- 2.1. The derivation of finite equations.- 2.2. Application to the heat equation.- 2.3. The “Hermitian” methods.- 2.4. An example.- § 3. Some theoretical aspects of the finite-difference methods.- 3.1. Choice of mesh widths.- 3.2. An error estimate for the inhomogeneous wave equation.- 3.3. The principle of the error estimate for more general problems with linear differential equations.- 3.4. A more general investigation of error propagation and “stability”.- 3.5. An example: The equation for the vibrations of a beam.- § 4. Partial differential equations of the first order in one dependent variable.- 4.1. Results of the theory in the general case.- 4.2. An example from the theory of glacier motion.- 4.3. Power series expansions.- 4.4. Application of the finite-difference method.- 4.5. Iterative methods.- 4.6. Application of Hermite’s formula.- § 5. The method of characteristics for systems of two differential equations of the first order.- 5.1. The characteristics.- 5.2. Consistency conditions.- 5.3. The method of characteristics.- 5.4. Example.- § 6. Supplements.- 6.1. Monotonic character of a wide class of initial-/boundary-value problems in non-linear parabolic differential equations.- 6.2. Estimation theorems for the solutions.- 6.3. Reduction to boundary-value problems.- 6.4. Miscellaneous exercises on Chapter IV.- 6.5. Solutions.- V Boundary-value problems in partial differential equations.- § 1. The ordinary finite-difference method.- 1.1. Description of the method.- 1.2. Linear elliptic differential equations of the second order.- 1.3. Principle of an error estimate for the finite-difference method.- 1.4. An error estimate for the iterative solution of the difference equations.- 1.5. Examples of the application of the ordinary finite-difference method.- I. A problem in plane potential flow.- II. An equation of more general type.- III. A differential equation of the fourth order.- 1.6. Relaxation with error estimation.- 1.7. Three independent variables (spatial problems).- 1.8. Arbitrary mesh systems.- 1.9. Solution of the difference equations by finite sums.- 1.10. Simplification of the calculation by decomposition of the finite-difference equations.- § 2. Refinements of the finite-difference method.- 2.1. The finite-difference method to a higher approximation in the general case.- 2.2. A general principle for error estimation.- 2.3. Derivation of finite expressions.- 2.4. Utilization of function values at exterior mesh points.- 2.5. Hermitian finite-difference methods (Mehrstellenverfahren).- 2.6. Examples of the use of Hermitian formulae.- 2.7. Triangular and hexagonal mesh systems.- 2.8. Applications to membrane and plate problems.- § 3. The boundary-maximum theorem and the bracketing of solutions.- 3.1. The general boundary-maximum theorem.- 3.2. General error estimation for the first boundary-value problem.- 3.3. Error estimation for the third boundary-value problem.- 3.4. Examples.- 3.5. Upper and lower bounds for solutions of the biharmonic equation.- § 4. Some general methods.- 4.1. Boundary-value problems of monotonic type for partial differential equations of the second and fourth orders.- 4.2. Error distribution principles. Boundary and interior collocation.- 4.3. The least-squares method as an interior and a boundary method.- I. Interior method.- II. Boundary method.- 4.4. Series solutions.- 4.5. Examples of the use of power series and related series.- 4.6. Eigenfunction expansions.- § 5. The Ritz method.- 5.1. The Ritz method for linear boundary-value problems of the second order.- 5.2. Discussion of various boundary conditions.- 5.3. A special class of boundary-value problems.- 5.4. Example.- 5.5. A differential equation of the fourth order.- 5.6. Direct proof of two minimum principles for a biharmonic boundary-value problem.- 5.7. More than two independent variables.- 5.8. Special cases.- 5.9. The mixed Ritz expression.- § 6. The Trefftz method.- 6.1. Derivation of the Trefftz equations.- 6.2. A maximum property.- 6.3. Special case of the potential equation.- 6.4. More than two independent variables.- 6.5. Example.- 6.6. Generalization to the second and third boundary-value problems.- 6.7. Miscellaneous exercises on Chapter V.- 6.8. Solutions.- VI Integral and functional equations.- § 1. General methods for integral equations.- 1.1. Definitions.- 1.2. Replacement of the integrals by finite sums.- 1.3. Examples.- I. Inhomogeneous linear integral equation of the second kind.- II. An eigenvalue problem.- III. An eigenvalue problem for a function of two independent variables.- IV. A non-linear integral equation.- 1.4. The iteration method.- 1.5. Examples of the iteration method.- I. An eigenvalue problem.- II. A non-linear integral equation.- III. An error estimate for a non-linear equation.- 1.6. Error distribution principles.- 1.7. Connection with variational problems.- 1.8. Integro-differential equations and variational problems.- 1.9. Series solutions.- 1.10. Examples.- I. An inhomogeneous integro-differential equation.- II. A non-linear integral equation.- § 2. Some special methods for linear integral equations.- 2.1. Approximation of kernels by degenerate kernels.- 2.2. Example.- 2.3. The iteration method for eigenvalue problems.- § 3. Singular integral equations.- 3.1. Smoothing of the kernel.- 3.2. Singular equations with Cauchy-type integrals.- 3.3. Closed-form solutions.- 3.4. Approximation of the kernel by degenerate kernels.- § 4. Volterra integral equations.- 4.1. Preliminary remarks.- 4.2. Step-by-step numerical solution.- 4.3. Method of successive approximations (iteration method).- 4.4. Power series solutions.- § 5. Functional equations.- 5.1. Examples of functional equations.- 5.2. Examples of analytic, continuous and discontinuous solutions of functional equations.- 5.3. Example of a functional-differential equation from mechanics.- 5.4. Miscellaneous exercises on Chapter VI.- 5.5. Solutions.- Table III. Finite-difference expressions for ordinary differential equations.- Table IV. Euler expressions for functions of one independent variable.- Table V. Euler expressions for functions of two independent variables.- Table VII. Catalogue of examples treated.- Author index.