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This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples. Exercises and student projects are available on the book's webpage, along with Matlab mfiles for implementing methods. Readers will gain an understanding of the essential ideas that underlie the development, analysis, and practical use of finite difference methods as well as the key concepts of stability theory, their relation to one another, and their practical implications. The author provides a foundation from which students can approach more advanced topics.

Finite differences --- Differential equations --- Différences finies --- Equations différentielles --- Finite differences. --- Differential equations. --- Basic Sciences. Mathematics --- Differential and Integral Equations --- 519.62 --- 519.63 --- 681.3*G18 --- 517.91 --- 517.95 --- Differences, Finite --- Finite difference method --- Numerical analysis --- 517.95 Partial differential equations --- Partial differential equations --- 517.91 Ordinary differential equations: general theory --- Ordinary differential equations: general theory --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 519.62 Numerical methods for solution of ordinary differential equations --- Numerical methods for solution of ordinary differential equations --- 517.91 Differential equations --- Differential and Integral Equations. --- Différences finies --- Equations différentielles --- 517.91. --- 681.3 *G18 --- Numerical solutions --- Différences finies. --- Équations différentielles.