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I am looking for an introductory book on Numerical Analysis and Differential Equations. I have done my B.Sc. in Math and I'm preparing for M.Sc entrance exams. The syllabus for the exam contains the following topics:

  • Existence and Uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ordinary differential equations, System of first order ordinary differential equations, General theory of homogeneous and non- homogeneous linear ordinary differential equations, Variation of parameters, Sturm Liouville boundary value problem, Green’s function.
  • Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs, Classification of second order PDEs, General solution of higher order PDEs with constant coefficients, Method of separation of variables for laplace. Heat and Wave equation.
  • Numerical solutions of algebraic equation, Method of iteration and Newton-Raphson method, Rate of convergence, Solution of systems of linear algebraic equations using Guass elimination and Guass-Seidel method, Finite differences, Lagrange, Hermite and Spline interpolation, Numerical integration, Numerical solutions of ODEs using Picard, Euler, modified Euler and second order Runge- Kutta methods.

I've had an introduction to ODE from Pollard and Tenenbaum but I have no prior experience with PDE and Numerical Analysis. This exam only contains MCQ so I am more interested in getting to know how to apply a particular method fast, rather than the proofs and justifications behind it, something like Stewart's Calculus that has lots of solved problems, skips the harder proofs and is very concise and suitable for self study.

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  • $\begingroup$ The proposed answer contains some excellent references. I would second all of the books by A.Quarteroni for everything related to the numerical analysis of differential problems. I would only add the book by Lawrence C. Evans: Partial Differential Equations, for your second topic. The first part of the book contains most of what you might need. $\endgroup$ – bgsk Mar 27 at 7:12
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My favorite books for these three topics are

  1. Hairer Norsett Wanner, Solving Ordinary Differential Equations I - NonstiffProblems
  2. Salsa, Partial Differential Equations in Action
  3. Quarteroni, Sacco, Saleri, Numerical Mathematics

In particular, the first reference is a classic in numerical integration of ODEs but contains a detailed introductory chapter on the basic theoretical aspects of ODEs. The second is a very well-written introductory book on PDE theory. While the first chapters provide with practical methods for solving one-dimensional PDEs in simple cases (separation of variables etc), the author did not forget some interesting theoretical aspects about PDEs. The third is a classic introduction on Numerical Analysis, and treats, among other topics, linear systems (direct and iterative methods), interpolation, numerical integration and basics on the integration of ODEs. All books contain several examples and interesting exercises.

I'm not aware of the existence of a single book treating all of the above topics.

PS: I'm not payed by Springer for advertising their books :)

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