Numerical Analysis and Differential equations book recommendations focusing on the given topics. 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:


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*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.
 A: My favorite books for these three topics are

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*Hairer Norsett Wanner, Solving Ordinary Differential Equations I -
NonstiffProblems

*Salsa, Partial Differential Equations in Action

*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 :)
A: In my opinion, the best books on analytical solutions of ODEs are:

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*Differential Equations with Applications and Historical Notes
By George F. Simmons: if you're interested in the engineering approach. It lists all the various cases and types of Differential Equations you can encounter in physics along with methods to solve them

*Ordinary Differential Equations. Authors: Arnold, Vladimir I.: still the best book out there in my opinion. The mathematical approach. It doesn't teach you just a bunch of techniques on how to solve the equations, it teaches instead what a differential equation is, how to understand if a solution exists, if it's unique, how to interpret an equation qualitatively from its phase space before even solving it, and a lot of other interesting theorems

The best books on numerical solutions of ODEs are:

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*Introduction to Numerical Analysis. Authors: Stoer, Josef, Bulirsch, R: excluding PDEs, this is the Bible of everything numerical


*An Introduction to Numerical Analysis. Endre Süli, University of Oxford, David F. Mayers, University of Oxford: this is the best one to pick-up to self teach numerical calculus, because it manages to explain the concepts clearly but keep the mathematical rigor at the same time
The best books on analytical solutions of PDEs out there are:

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*Partial Differential Equations: Second Edition by Lawrence C.
Evans: the Bible on PDEs

*Partial Differential Equations: An Introduction, 2nd Edition by Walter A. Strauss: really good introduction

*Partial Differential Equations in Action by Sandro Salsa: another really good one (it's cheaper than the other two)

The best books on numerical solutions are:

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*Numerical Models for Differential Problems Authors: Quarteroni, Alfio

*Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation. by Grégoire Allaire and Alan Craig
