14. Digital Differential Equations

It is discussed the similarities and differences of the difference equations and digital differential equations (the later have two degrees of precision while the former only one).


APPENDIX

Here are some new methodologies of solving ordinary and partial differential equations within the new axiomatic digital mathematics based on the concept of resolution which is deeper in mathematical ontology than just a grid of points in numerical methods of ODE and PDE. We apply it as example on solving the partial differential equations of Navier-Stokes equations of compressible Newtonian fluids. We deviate from our standard approach to be always only inside the new axiomatic digital mathematics, and we create new techniques within the classical mathematics with the infinite inspired by the techniques of the axiomatic digital mathematics. 

Creating 3D mental images with the imagination

The physiscists andd nobel prize winner R Feynamnn often was explaining that he was training his imagination imagining in 3D mental images the moving molecules of a gas in  box, and perceiving that temperature as their avearge velocity, the ressure as the average moemntum of the walls of the box, and the density as the average number of particles per unit of volume, He was explaing that he was getting much pleasure from this spiritual excerise of craeting 3D mental images and simulating the reality. 

Excatly this 3D VISUAL MENTAL IMAGES LANGUAGE parctice is  UTILIZED BY ME AND  is  required by someone to understand and develop the digital or natural mathematics, in geometry, real numbers and analysis.

Rules for phantasy and drawing of figures.

As initially we considered a system of digital real numbers R(m,n,p,q) we consider the points of P(m), P(n) as visible in the figures while the points of P(p) as invisble pixels , and those of  P(q) as invisible atoms. Therefore, even the points and seemingly infinitesimals that will be defined below, of P(n) relative to P(m) are considered as visible. This is in accordance with the habit in classical mathematics to make the points visible, although they claim that they have zero size.

THE RESOLUTIONS METHODOLOGY IN FINDING AND STUDYING SOLUTIONS OF SYSTEMS OF DIFFERENTIAL EQUATIONS (E.G.  NAVIER-STOKES PARTIAL DIFFERENTIAL  EQUATIONS).

1)The finite resolutions and their triple equality of quantities (ε, δ, α) , =1, =2, =3 .

(The definition of the finite resolution does no include the intermediate points inside its cuboids. It is a finite set of points, a double lattice, a double grid. But we give a separate definition for infinite resolution when the space-time is unbounded. We prefer to work the NsE project on the periodic formulation, on the compact 3-dimensional torus T3. We give also symbolization of the finite resolution, based on the decimal representation of the real numbers thus of R^n that makes it clear, and also clear the relative distance of the unit pf space, from the minimum pixel size of minimum invisible cuboids=pixels . The mid (double) precision cuboids of a resolution may be already invisible, as elementary volumes, and inside the visible points (low precision) , while the pixels are inside the mid-precision cuboids and are the high (triple) precision. Below we shall see that the high-precision pixels will represent the particles of the fluid. While, the mid-precision cuboids the non-moving elementary volumes on which we observe and state the equations of a fluid. The differences of the values of the magnitudes of the fluid on the adjacent mid-precision elementary volumes, are already below the single precision of the measurement magnitude of the fluid that is the single (low) precision. The single or low precision defines the visible points of the fluid. The bins (ε, δ, α) refer to the visibe points invisble elementary volumes and particle size. The (ε, δ, α) is called the signature of the resolution.There is a natura ordering of resolutions R1<R2 if for their signatures holds (ε1, δ1, α1) <=(ε2, δ,2 α2) respectivelly).

2) Existence or definition of functions on finite resolutions

(Every classical function on R^n space by restriction (inheritance) defines a function finite or infinite resolution. On many resolutions many functions. Conversely we give conditions, that functions on all finite or infinite resolutions as an inductive system define a unique function of R^n or torus T3)

3)Equality of functions on finite resolutions

(Two functions of R^n or torus T3, equal on all resolutions (infinite or finite respectively) are equal functions on R^n or torus T3)

3. Bottom-up inheritance of functions from a resolution to coarser resolution by averaging or interpolation (Zoom-out).
   Here only the e is increased from the signature (e,d,a). That is only the visible points cuboids are increased not the elementary volumes or particles.
4. Top-down extension of a function from a resolution to a finer resolution (Zoom-in). Here both the e and d is decreased from the signature (e,d,a). That is the visible points cuboids and the elementary volumes are decreased in size, are refined, not the particles.

5)The continuity on finite resolutions

(The continuity on a resolution is defined by the pixel (infinitesimals) of double decimal precision , as the Leibnitz continuity. No limits. Although the ε-δ definition of Weierstrass does apply very well.)

6)Derivative on finite resolutions

(Similar the definition of derivative and partial derivatives as quotients of double decimal precision pixels (infinitesimals), like the Leibnitz way. No limits)

7)Integration on finite resolution.

( The integral on a finite resolution is simply a finite sum, and double decimal precisions pixels (infinitesimals) are involved)

8)The smoothness on finite resolution

(Smoothness is defined on finite resolution with partial derivatives )

9) The physical natural real minimal resolution R(ε0, δ0, α0).

For all physical applications of the pfuid dynamics it is required the minimal physical resolution. Hyper-Continuity is referring to this as well as Hyper-smoothness.

10) The Cauchy-Kovalevskaya theorem of local existence and uniqueness of solutions and the resolution-NSE (on a finite resolution).

Since the equality on a finite resolutions is only single-precision on it, the resolution-NSE has

analytic coefficients, as the Taylor expansion becomes zero after a finite number of terms. Therefore the C-K local existence-uniqueness for quasilinear systems of PDE, applies provided we extent the NSE that are quadratic PDE to linear PDE with additional variables. But as we shall see later, we can do better and derive existence and uniqueness of global solutions too.

11) Definition of functions in the classical sense from that of an inductive system of resolutions larger than the minimal physical.

(We give conditions, that functions on all finite or infinite resolutions as an inductive system define a unique function of R^n or torus T3)

12) Characterization of equality of functions in the classical sense from that of an inductive system of finite resolutions larger than the minimal physical.

(Two functions of R^n or torus T3, equal on all resolutions (infinite or finite respectively) are equal functions on R^n or torus T3)

13) Characterization of classical continuity from an inductive system of resolutions larger than the minimal physical. Classical continuity means also continuity on the minimal physical resolution.

(Functions on R^n or T3, that when restricted on any resolution are continuous, are also continuous in the classical calculus sense. The e-d of the Weierstrass apply here. If a function is continuous in the classical sense then the for every e, there is e-1^st precision resolution with d-2^nd precision such that the inherited function on the resolution r is resolution-continuous. Conversely if the function is discontinuous some where say at a, the there is a resolution with e-1st-precison, such that for any d-2nd-precision the inherited function is resolution-discontinuous. NEVERTHELESS DOES NOT MEAN THAT IF WE HAVE A DISCONTINUOUS IN THE CLASSICAL SENSE FUNCTION, THERE IS NO RESOLUTION SUCH THAT THE INHERITIED FUNCTION IS NOT CONTINUOUS! THERE MAY EXIST A RESOLUTION THAT THE INHERITED FUNCTION (OF THE CLASSICAL DISCONTINUOUS FUNCTION) IS RESOLUTION-CONTINUOUS! ).

14) Characterization of classical smoothness with smoothness from an inductive system of resolutions larger than the minimal physical resolution.Classical smoothness means also smoothness on the minimal physical resolution.

(Functions on R^n or T3, that when restricted on any resolution are smooth , are also in the smooth in the classical calculus sense )

15) Integration in the classical sense from that of an inductive system of resolutions larger than the minimal physical resolution. Integration is also a simple summation on the minimal physical resolution.

(For a Lebesgue and Riemann integrable function on R^n or T3, the integral can be calculated, from the corresponding integrals of the restriction functions on all resolutions)

16) The NSE on finite resolutions and the minimal physical resolution. The resolution-NSE.

The main difficulty  is that the general methods of solutions of the NSE so far, do not involve a systematic theoretical assessment of them, so that general properties of the solutions can be deduced. The numerical methods suffer by that convergence is intervened. While non-numerical methods are not general enough to cover , the general case of the NSE. That is why, the current methodology with the resolutions is the appropriate. We are not concerned with convergence instead we are based on the inductive relation of resolution-solutions and general solutions.

( On a finite resolution, the NSE from the species of PDE, goes to the species of systems of algebraic quadratic equations, through the pixel-deltas. We call it resolution-NSE. The elementary volumes may be interpreted as the single precision cuboids of the resolution where density and pressure of the double-precision cuboids or pixels within the single precision cuboids. In fact we find here 3 different lattices, or grids. The finite, the mid( double)-precision pixel cuboid volumes and the high (triple) precision cuboid volumes.

The high (triple)-precision pixels will represent the particles of the fluid. While, the mid (double)-precision cuboids the non-moving elementary volumes on which we observe and state the equations of a fluid. The differences of the values of the magnitudes of the fluid on the adjacent mid-precision elementary volumes, are double precision magnitudes and already below the single precision of the measurement magnitude of the fluid that is the single (low) precision. The single or low precision defines the visible points of the fluid.

Then the resolution-NSE are simply the ( energy and ) momentum conservation of the single-precision cuboids as non-moving spatial observatory cuboid-windows with the double precision cuboid volumes as particles of the fluids. In other words the elementary-volumes are not parts of the fluid are non-moving abstract special cuboid containers to measure quantities of the fluid the passes through them. The pixel-cuboids of triple (high) precision represent particles of the fluid. The density represents number of such particles in the elementary volume. The pressure, at a wall of the elementary volume , measures the vertical to the wall momentum of each particle that crosses the imaginary wall. Although the density may me constant (incompressibility) the pressure may change, for a single elementary cuboid volume, the vertical momentum crossed at different walls may be different. We must remember that the quantities on the elementary volumes are averages on the pixel-particles. The velocity at the elementary volume is the average sample velocity of the velocities of the particles. And so is the temperature as average norm of velocity of the particles, or the internal energy of the fluid at the elementary cuboid volume. Similarly for the viscosity coefficient and external body-volume forces ad that elementary volume.

Conversely solutions of such resolution-NSE on all resolution, with appropriate inductive compatibility conditions, give solutions of the original NSE. Existence , uniqueness, and smoothness are preserved. Also non-smoothness and finite blow-up times , studied by the species of algebraic quadratic equations, are preserved after the inductive compatibility conditions. We restrict to the 2-dimensional case so that we can compare with the 3-dimesnional case. )

17) The energy of the resolution, in resolution-NSE0.

We may apply ,methods of energy for the velocity generic particle of the normal digital particle fluid, and try to prove that it is impossible , that the velocity will converge to infinite in a finite time, thus refuting the classical result of finite Blow-up time. In this we must take in to indispensable account and be based critically on the non-compressibility at least in the form of constant density, as well as the smoothness of the pressure. We may derive from this some regularity of the statistics. Also we may change the definition of the normal digital particle fluid , so as to have simple proof of the above reduction to absurd.

18) The trajectory path of an elementary volume of the resolution, at a resolution-solution of the resolution-NSE.

19) Local in time existence of resolution-smooth solutions of the resolution-NSE, by the corresponding results in the classical non-resolution setting.

20) Existence in 2-dimensions of global in time resolution-smooth solutions of the resolution-NSE , by the corresponding Ladyzhenskaya result.

In 2-dimensions the phenomenon of vortex axial stretching and split of the vortex to smaller and smaller vortexes and eddies cannot take place. It is a 3-dimensional phenomenon. Still this 3-dimensional mechanism in producing a spectrum of eddies or whirls (the smaller the higher the spin) that traditionally is considered turbulent flow, could still be described with smooth solutions of the NSE ! The Kolmogorov length scale of turbulence indicates that even in turbulence there is a un upper bound to the scale of eddies and an upper bound to the vorticities and their speed norm and thus they do not have the effect of the velocities converging to infinite, as time approaches a particular moment (at the finite Blow-up time) ! 

Of course there are turbulent flows, that may create singularities of very small eddies in other words point of non-smoothness of the flow.

21) The NSE on finite resolutions and systems of intersecting (parabolic) Quadratic equations.

(We apply here the theory of quadratic surfaces, and their normal forms. In particular we prove that the equations of the resolution-NSE are hyperbolic type quadratic equations. )

22) Existence of resolution-NSE resolution-solutions on finite resolution

(Itis proved that the resolution-NSE do have solutions given the initial boundary resolution conditions. We apply the resolution-NSE recursively on all the points of the time/space resolution and we get a finite system of algebraic equations that we call resolution-solution of the resolution-NSE. They represent the energy , mass or number of particles and momentum conservation of finite many pixel cuboids of the resolution. The elementary single precision volumes are simply cuboids that through their magnitudes like density, pressure , velocity etc describe the statistical movements of the double precision pixel cuboids or particles. And the resolution-NSE is simply in words density times (time-acceleration+space-acceleration) of the elemtary-volumes equals force as pressure space-change on elementary volumes and viscosity loss of momentum proportional to second derivatives of the velocity.

Here the relation of Du/dt=∂u/∂t +u(inverse∆)u is utilized (derivative rule of product and composite)

Conservation of the mass or number of particles (continuity equation)

All of these magnitudes as sample-group averages of the sample of pixel-particles in the elementary-volumes of the fluid. The initial Newton’s equation on an elementary volume is a solvable differential equation, and so should be the NSE, that is on a lattice of such elementary volumes. For the existence we start with resolution of minimal number of points we verify the existence and then we utilize induction on the points pf the finite space-time resolution. After all it represent motion of incompressible fluids with energy and momentum conservation. And in out formulation we directly represent the physical atoms of the fluid (as almost incompressible crystal fluid) and volumes of them. If anything must be added to the formulation of the energy and momentum conservation beyond the resolution-NSE we do formulate it, so as derive exactly natures existence and uniqueness of motion of fluids. Incompressibility has a nice interpretation on resolutions as slight compressibility that is not more than double precision pixel magnitude. We restrict to the 2-dimensional case so that we can compare with the 3-dimesnional case. )

23) Uniqueness of solutions of resolution-NSE on finite resolutions

(It is proved that the resolution-NSE not only do have resolution-solutions given the initial boundary resolution conditions but the resolution-solution is unique. After all it represent motion of incompressible fluids with energy and momentum conservation. Incompressibility has a nice interpretation on resolutions as slight compressibility that is not more than double precision pixel magnitude. We restrict to the 2-dimensional case so that we can compare with the 3-dimensional case. )

24) Local smoothness of resolution-solution of the resolution-NSE on finite resolutions (From the previous local classical smoothness of the resolution-solution of the resolution-NSE, we derive local resolution-smoothness of the resolution-solutions of the resolution-NSE. Also a smooth change in the initial conditions will result in local smooth change in the subsequent resolution-solutions. Resolution-solutions of the resolution-NSE evolve smoothly in time)

25) Alternative method: Digital real fluids observable through a finite resolution

26) The resolution-NSE of a digital real fluid

27) Existence of a normal digital real fluid, for the initial (boundary) conditions of the resolution-NSE.

Here we take a deterministic real fluid of equal size particles at pixels of the resolution, and at each elementary cuboid volume inherit, the velocity of the volume, while when it changes elementary volume it experiences a deceleration according to the NSE0 formula of viscosity, while it takes also the other elementary volume velocity of new time step of an implicit defined resolution-solution or keeps the same as alternative rule. The later rule is according to the Newton law and momentum conservation, the first not! But both rules give particle-fluid satisfying the resolutions-NSE0. The former requires to already know one, the later not, and can be used to define one! As time goes on in discrete steps we propagate the motion of particles as of constant velocity along their straight line linear trajectories inside their elementary volumes. (unless they collide or change elementary volumes).

28) The normal digital real fluid evolves locally continuously and smoothly in time.

All of these changes are at double precision, therefore are continuous and smooth at the single precision. Also when particles collide the collision is perfect elastic, and they exchange velocity, but as it is equal as they are in the same volume, it is as if each particle goes through the other without colliding. In case they collide, at the walls of the elementary volumes, again, they exchange velocities, but as the elementary volumes are adjacent again it is a change at double precision an zero at single precision. Therefore we may conclude that the velocities of the particles of the normal real fluid, are continuous in time, and so is the pressure, due to the statistical interpretation of the pressure! Finally as the new velocities of the normal real fluid are averages of the velocities of the particles, we conclude that the field of velocities of the normal real fluid (not that of the particles) is continuous and smooth (1^st derivative).

29) Derivation of a resolution-solution of the resolution-NSE, by the evolution of a digital real fluid.

Here we define the elementary volumes velocities and wall-pressures, and densities by the statistical average quantities of the real fluid. Then we repeat the argument as in the derivation of the NSE0, but this time on the normal real fluid. As it is a real fluid that its particles follow the law of motion of Newton, so it happens with the elementary volumes, and therefore the normal real fluid satisfies the resolution-NSE0.

We may mention here the publication of A. Muriel http://arxiv.org/ftp/arxiv/papers/1011/1011.6630.pdf

30) Existence and uniqueness of solutions of NSE from that on all resolutions.

(We prove that existence and uniqueness of solutions of the resolution-NSE on all resolutions, derive existence and uniqueness of the solutions of NSE.)

References

W. Hurevicz

Lectures on ordinary Differential Equations

At the end of this chapter, there is 

a) Advantages-disadvantages of these new digital mathematics compared to the classical analogue, infinitary mathematics. 

b) A fictional discussion in dialogue form of celebrated historic creators of the old mathematics praising or questioning the new mathematics compared to the old.