Books and resources on PDEs that use Mathematica and Matlab

For numerical analysis, I can recommend:

• Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems by R. J. LeVeque.
• Spectral methods with MATLAB by L. N. Trefethen.

Both use MATLAB.

4 reference books for the study of PDE with MATLAB:

Coleman, Matthew P. An introduction to partial differential equations with MATLAB. Second edition. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. CRC Press, Boca Raton, FL, 2013.

Thorough treatment of PDEs and their applications, includes numerous problem-solving exercises, MATLAB code on the author's website, first edition from 2005.

Li, Jichun; Chen, Yi-Tung Computational partial differential equations using MATLAB. With 1 CD-ROM (Windows, Macintosh and UNIX). Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. CRC Press, Boca Raton, FL, 2009.

Provides standard finite difference and finite elements, novel techniques, such as high-order compact finite difference and meshless methods, applications from the fields of mechanical and electrical engineering, presents both theoretical numerical analysis and practical implementations, many computer projects and problems, includes a CD-ROM with MATLAB source code.

Stanoyevitch, Alexander Introduction to numerical ordinary and partial differential equations using MATLAB. Pure and Applied Mathematics (New York). Wiley-Interscience, Hoboken, NJ, 2005.

Thorough coverage of analytic concepts, geometric concepts, programs and algorithms, and applications. Extensive chapter on the finite element, including mesh generation, enables one to numerically solve general elliptic boundary value problems, FTP site that includes downloadable files.

Cooper, Jeffery Introduction to partial differential equations with MATLAB. Applied and Numerical Harmonic Analysis. Birkhauser Boston, Inc., Boston, MA, 1998.

Includes analytical and numerical methods, separation of variables technique, together with transform methods associated with Fourier, Green's function, variational methods, theory and applications of finite elements for boundary value problems, method of finite differences is also considered.

and three books for PDE with Mathematica:

Adzievski, Kuzman; Siddiqi, Abul Hasan Introduction to partial differential equations for scientists and engineers using Mathematica. CRC Press, Boca Raton, FL, 2014.

Provides fundamental concepts, ideas, and terminology related to PDEs, discusses separation of variable method, studies the solution of the heat equation using Fourier and Laplace transforms, examines the Laplace and Poisson equations of different rectangular circular domains, and discuss finite difference methods.

Kythe, Prem K.; Puri, Pratap; Schäferkotter, Michael R. Partial differential equations and boundary value problems with Mathematica. Second edition. Chapman & Hall/CRC, Boca Raton, FL, 2003.

Theory and applications for solving initial and boundary value problems involving, in general, the first-order partial differential equations, and in particular, the second-order partial differential equations of mathematical physics and continuum mechanics.

Vvedensky, Dimitri Partial differential equations with Mathematica. Physics Series. Addison-Wesley Publishing Company, Wokingham, 1993.

Covers linear and nonlinear partial differential equations with exemplar examples, inspired by the symbolic software Mathematica.