This is a worthwhile question.

The relationship between PDEs and ODEs is given in detail in a bit hefty 2018 paper:

Finite PDEs and finite ODEs are isomorphic

https://arxiv.org/pdf/1807.09327.pdf

Yes, it related to the location of the variables which can affect the values of the other variable in time. In our universe, space and time evolution is one of the important parameters. Modeling through PDE is to study the space and time factor role on others paraments

@Ram Singh thank you for your answer.

So what are the reasons available for demonstrating this hypothesis?

@Luis - Felipe Velázquez - León

Thank you for your answer, but my question is: when we use ode equations in modeling, what is justification for when we add pde to that, and it shows the spatiotemporal behavior?

This is a worthwhile question.

The relationship between PDEs and ODEs is given in detail in a bit hefty 2018 paper:

Finite PDEs and finite ODEs are isomorphic

https://arxiv.org/pdf/1807.09327.pdf

Dear Yeganeh Shaverdi:

Is your question motivated by situations such as the original Turing mathematical model of morphogenesis using a coupled system of ordinary differential equations, from one side, and its formulation by two partial differential equations, from the other side?

Hello,

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Yeganeh Shaverdi - it would be much profitable for you to give a specific case of, as you say, using PDE in solving ODE since generally speaking one does not solve something via other, more complicated.

In particular, the multiple scales method is used to analyze nonlinear systems presented by ODEs. This method is realized by decision of PDEs. It permits to describe a behaviour of the system under consideration with different time scales, including fast and slow time scales.

Interesting answers from different scholars from their point of views. Upcoming researchers would really be grateful for this knowledge shared

The relationships between ODEs and PDEs were a classic subject in mathematics in the 19th century. The most significant contributions are credited to G. Monge, F. Klein, and M.S. Lie. For biographical notes giving the excellent description of their works, please consult

http://www-groups.dcs.st-and.ac.uk/history/Biographies/Monge.html

http://www-history.mcs.st-andrews.ac.uk/Biographies/Klein.html

http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html

The best-known theory, which describes basic connections between PDEs and ODEs, was developed by Monge for the first-order PDEs. For the description of the theory you can consult the attached article

In the area of applied sciences the PDE’s are often converted to a large set of the ODE’s by applying the Method of Lines which is used in the numerical solution. The concise description is presented in Wikipedia:

https://en.m.wikipedia.org/wiki/Method_of_lines