References on computational PDE (in fluid dynamics, solid mechanics, etc) that emphasize both rigorous analysis and coding

Question

I'm interested in learning about computational aspects of PDE and integro partial differential equations.

In particular, I'd like to know some reference monographs that cover PDE and IPDE from in computational physics, computational fluid dynamics, computational solid mechanics, computational biology (including biomodeling, genomics, neuroscience), computational chemistry and pharmacology.

I'd appreciate both rigorous mathematical and numerical analysis and extensive use of sample code written in MATLAB/Mathematica.

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Here are two related questions that I have checked out:

• Reference request: numerical analysis of PDEs and integro partial differential equations
• Books and resources on PDEs that use Mathematica and Matlab