Fellow researcher, the short answer is no.

"the existence and the uniqueness of the solution are ensured" ? This is wrong.

The disciplines, although intertwined, makes completely different assumptions and present completely different methods and procedures. Well-structured problems and math require a body of absolute assumptions (postulates) that allows you to reach a univocal solution and that's it. You answers, procedures, insights, all come from the nature of the postulates and nothing more. In physics, the postulates come from experimentation and application of the scientific method. When you trust enough your postulates, you make a physical theory using math based on your postulates. Is this theory wrong? Do experiments to know. If it is, it may be due wrong mathematical approach or because the postulates are wrong, but physics have this nice feature to always come back to experimentation. For example, energy and momentum are conserved in the absence of dissipation right? But only this is not enough to calculate motion equations, but what else do you need? Nothing more: use Newton physics. The particles present to big De Broglie wave length? Use Quantum Mechanics. The particles are too massive or too fast? Use Einstein physics. Note that the answers are completely different if your observations are different. In math, in principle, you can formulate all possible sets of postulates and make all possible theories about how the universe works (or can possible work), but math does not have the tools to select which theory describes the world or if it is "right" or ''wrong'' in the sense of observations.

I hope this discussion was helpful.

Physics is a mathematics-based discipline. It is not a special case of mathematics. They are different disciplines.

In general the answer is no because physics is an experimental science and mathematics is not. Best wishes David Booth

[Is it true that physics is a special case of mathematics?]

First of all, I must thank my colleagues for their helpful answers.

Obviously, we consider theoretical physics since experimental physics with its measurements in physical SI units and dimensions goes beyond the scope of mathematics,

A particular case in the broad sense means that each element exists in the general case or in the mathematical language, it is a subset of the general case. Therefore, a somewhat clearer wording of the question,

[Is it true that theoretical physics is a subset of mathematics?]

I guess so, it's true that physics is a subset of math.

I also assume that the problem of the existence and uniqueness of the solution is the one that is outside the domain where physics and mathematics meet since in physics the existence and uniqueness of the solution solution are ensured whereas in mathematics, it is the general case of physics, the existence and the unicity of the solution pass before all.

An important case is the famous partial heat diffusion equation conduction/diffusion in 2D and 3D where the solution exists but is neither analytically nor numerically accessible. The same equation can be easily solved in 2D and 3D time-dependent geometry for any arbitrary boundary conditions and initial condition when we apply a physical transition matrix satisfying physical probability conditions on free and fixed nodes (Cairo modeling

techniqus).

Another more striking example is numerical double integration and triple integration called curve fitting or area/volume under the curve

The math offers much like trapezoidal and Simpson's rules, the former is accurate for linear fit (Y=aX+b) and the latter is accurate for quadrature fit (Z=Fn(X^2, Y ^ 2).

Again, the physical transition matrix satisfies physical probability conditions and offers more accurate solutions for quadrature, polynomials, and more complex functions such as Z = SQRT(XY)

Dude, no.

To divide physics in theoretical and practical is just a unfair abstraction. observation without theoretical investigation is just sight-seen. Theoretical investigation without experimental verification is just solving a 'puzzle' that any answer goes (no one can verify right?).... Do not divide those just to your argument to make sense. All you saying is "the math of physics is a subset of math" which is an empty logical sentence.

" I also assume that the problem of the existence and uniqueness of the solution is the one that is outside the domain where physics and mathematics meet since in physics the existence and uniqueness of the solution solution are ensured whereas in mathematics, it is the general case of physics, the existence and the unicity of the solution pass before all". Where in the world have you heard such an assumption? There is no such a thing as ensured existence and uniqueness of solutions in physics, in any field. The experimental investigation people carry is to determine the limits of the theoretical calculations and to find what is missing in the current theories. I give you a good example. Information carrying signals can not move faster than light right? Two particles can communicate through a signal (existence) which presents a specific velocity that can not surpass light velocity. If the particles present non-even charge distribution (charge or higher dipole momentum) then the velocity of the signal is the light velocity in that medium (uniqueness) right? Well, explain the Bell's inequality then. All physical theories are wrong as they are just approximations, but physics, contrary to math, allows you to explore its own inconsistencies to make more accurate models and improve on its own mistakes. Math is like scrap metal and physics is the machine you build with it, including its purpose, sketch and design.

In Leon Lederman's wonderful book, "The God Particle" he comments about a pecking order in physics. It goes something like, "the experimental physicists only defer to the theoretical physicists who in turn only defer to the mathematician and the mathematician only defer to God..."

Physics is not mathematics. The role of physics like all sciences is to understand the universe. The theoretical physicists' role is to develop theories that explains all the experiments from the past plus allows predictions to be made to test the and potentially falsify the theory. The role of the experimental physics is to develop and perform those experiments. There is often as much deep mathematics used by an experimental physicists as a theoretical physicists.

The role of the mathematician is simple to expend the understanding of mathematics through rigorous logical reasoning. If a physicists or any other scientist finds these tools useful - then great.

Arguably the greatest theoretical physicists of the second half of the 20 Century was Richard Feynman. Here is what he said about physics.

"It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong."

I would suggest for a better understanding of physics - what it is and isn't - a read of "The God Particle" would be recommended

https://en.wikipedia.org/wiki/The_God_Particle_(book) .

Note that the existence,stabilty and uniqueness of solution is ensured for blue area.(PHYSICS)

Note that the existence and uniqueness of solution is NOT ensured for red area.(MATHEMATCS) and it comes first and before all.

Theoretical physics is a subset of mathematics and probability falls to the heart. When mathematics meets physics, that is, their intersection in the language of set theory, universal physical laws and mathematical axioms apply simultaneously where the beauty and elegance of their combination appear intensely .

The abstract definition of probability in mathematics is also well defined, but it allows probability to be an arbitrary real number element of [0,1], whether rational or irrational. The mthematical definition is inadequate and difference is enormous and the conclusions diverge widely. The question is who do we track and when?

The correct definition of numerical physical probability is not the definition of frequency of occurrence as many people might think. Recent papers define the physical transition probability as the elements bi,j of a statistical transition matrix where these elements must satisfy all relevant universal laws. Now the statistical transition matrix B is well defined and uniquely specified,

It follows that the probability must be a rational number and that real time is completely lost and transformed into dimensionless time, The universal laws of physics to follow are,

Conservation of total energy, Symmetry in 3D space (ai,j=aj,i), No preferable direction, Probability of all space is unity, Boundary conditions at boundaries are subject to the same probabilistic conditions at free nodes. ...etc.

Can this physical probability solve the Laplace-Poisson PDE in the general case without the need for the FDM technique? The answer is yes and published in several articles since 2020.

Can this physical probability solve the finite double integration or the subsoil volume for an arbitrary function Z=f(x,y) for any number of free nodes n? Below is the transfer matrix for the double integration of 9 free nodes on a square x=1,2,3 & y=1,2,3. The answer is yes and the resulting transfer matrix is ​​more accurate than trapezoidal rule and Simpson's rule.

1 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 0.44444444775581360 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 .. TOTALREX.. 1.00000000

2 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.44444444775581360 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 .. TOTALREX.. 1.00000000 3 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 0.44444444775581360 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 .. TOTALREX.. 1.00000000 4 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.44444444775581360 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 .. TOTALREX.. 1.00000000 5 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 0.44444444775581360 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 .. TOTALREX.. 1.00000000 6 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.44444444775581360 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 .. TOTALREX.. 1.00000000 7 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 0.44444444775581360 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 .. TOTALREX.. 1.00000000 8 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.44444444775581360 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 .. TOTALREX.. 1.00000000 9 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 0.44444444775581360 0.11111111193895340 2.7777777984738350E-002 0.11111111193895340 2.7777777984738350E-002 .. TOTALREX.. 1.00000000

Can this physical probability show that the infinite power series [(1+x)/2]^N is equal to (1+x)/(1-x), ∀x∈[0,1]. The answer is still yes. etc..etc.

Dear all, it's very important to remark that:

1. If a physicist has not knowledge in mathematics, his work do not signify something truthful. Physics without mathematics or maked ignoring math knowledge is only a story, often nonvalid!

2. A mathematician, knowing almost nothing about physics, can obtain remarcable scientific results in mathematics, without needing any kind of physics!

That's a great difference!