A novel *R-based perspective on solving ordinary differential equations
AA novel ∗ R -based perspective on solving ordinarydifferential equations M. WeberJuly 2020
Abstract
In order to define the derivative f (cid:48) ( x ) of a function f : R → R weneed to “know something” about f in the environment of x . However,when we apply numerical routines for solving initial value problems, wedo it vice versa: We use f (cid:48) ( x ) in order to “tell something” about f inthe environment of x . Although, it is a one-way street: If f (cid:48) ( x ) and f ( x ) are given, we can not determine f ( x ) at any point x ∈ R differentfrom x . That is one conceptual problem of solving differential equationsin numerical mathematics. In this article, we will present a numericalalgorithm to solve very simple initial value problems. However, the changeof paradigm is, that we will not “leave” the point x . Solving ordinarydifferential equations is like searching for “recipes” f . Instead of tryingto find these recipes for values x ∈ R , we will learn them from specialnumbers in the “monad” of x . Hyperreal numbers [2] will play a crucial role in this article. ∗ R is the setof hyperreal numbers. The ring of infinitesimal numbers will be denoted as ∗ R i ⊂ ∗ R and the subset of finite numbers will be denoted as ∗ R f ⊂ ∗ R .Solving differential equations is usually understood as the searching for adifferentiable function f which solves a certain equation. What does “a functionis differentiable” mean? In non-standard analysis the derivative of a function f : R → R is usually defined as [2, 5]: dfdx ( x ) = st (cid:16) f ( x + dx ) − f ( x ) dx (cid:17) , (1)where dx is an infinitesimal number. The mapping st ( · ) is used to transformthe fraction in (1), which is a hyperreal number, to a real number (the “closest”real number). The standard part st ( · ) is a piecewise constant function in termsof hyperreal numbers, which is not defined for infinite numbers. If the standardpart of the fraction (1) exists and is independent from the choice of dx ∈ ∗ R i ,1 a r X i v : . [ m a t h . C A ] J u l eal axis the point x =0 ( ) is a root of F( ) f( ( )) is the „function“ that leads to F=0 - monad of x =0 - different infinitesimal numbers - classes of sequences converging to zero defines F( ) F( ) ( ) f( ( )) Figure 1: This sketch shows how we want to “not leave” the point x on the realaxis when solving ordinary differential equations F . Instead on “walking” on thereal axis, we take infinitesimal numbers into account, which can be visualizedas sequences which converge to zero. The algebraic relations between differentinfinitesimal numbers will provide the recipes to solve ordinary differential equa-tions. The whole approach starts with fixing one infinitesimal number α whichis then used to define derivatives and differential equations F . The solutions β of F are the roots of F . f ( β ) provides the recipe to compute the solutionfunction.then f is said to be differentiable. In this context it is important that the func-tion f is at least continuous in x , which equivalently means that the nominator f ( x + dx ) − f ( x ) is an infinitesimal number (otherwise, the fraction would befor sure infinite). Thus, for continuous functions the fraction is defined in theset of hyperreal numbers, but is not necessarily finite or independent from dx .This is similar to standard analysis, where “being independent from the choiceof the sequence dx → f : R → R exist,which are not differentiable at any point x ∈ R . One example - the Bolzanofunction - can be found in the script “Functionenlehre” of Bernhard Bolzano(arround 1830) and is available in [7]. These are examples for functions wherethe fraction in (1) can be computed for every dx , but the standard part doesnot exist or is not independent from the choice of dx . Alternative point of view.
One possibility to change the point of view isdiscussed (on a very basic level) in this article. If we leave out the calculation of2 t ( · ) in (1), then the solution of a differential equation would become dependenton dx . Some mathematical proofs are also constructed in such a way: We firstchoose an arbitrary but fixed number (here dx ) for which we do algebraic trans-formations, and only at the very end of our proof we consider the “structural” dx -dependence of our statement. Perhaps the structure of this dx -dependenceis the key to find a solution of the differential equation? Moreover, there issome other change connected to this approach. By leaving out the piece-wiseconstant function st ( · ) we can formulate the whole differential equation in termsof hyperreal numbers x (instead of real numbers x ) and even can get “closer” tolocality, because differentiation is a very “localized” process and the transitionfrom localized differentiation to a globalized solution of a differential equationwill have to be re-discussed as well. Generalizations.
Our approach uses hyperreal numbers. Other approachestry an analysis in terms of the even larger class of surreal numbers [8], where theconcept of “limits” of sequences is transferred to surreal numbers. In this articlewe will go back to the 17th and 18th century mathematics of algebraic analysis[1, 3] (before “limits” have been defined), but we will make use of hyperrealnumbers [2]. We will generalize the definition of derivatives to complex functions f : C → C by allowing for dx ∈ ∗ R i [i], where i is the imaginary unit. In thiscase, st ( · ) has to be applied to the real and complex part separately. The term“infinitesimal number” now also applies for elements of ∗ R i [i]. The term “finitenumbers” is used for ∗ R f [i]. Every number in ∗ R [ i ] which is not finite is denotedas “infinite”. Solving a first-order ordinary differential equation is the same like searching forthe zeros of a function F ( dfdx , f, x ). If there is a function f such that F ( dfdx , f, x ) =0 for all x , then f is called solution of the differential equation [10]. In thisarticle, we will restrict ourselves to differential equations which are given bypolynomials F ∈ C [ Z, Y, X ]. So far, the concept for checking whether f solves F is like this: First, we have to apply the definition (1) for the computation ofa real-valued dfdx and then we can check for F = 0. Alternatively, we can alsodefine the derivative of the function f to be f (cid:48) ( x ) = f ( x + α ) − f ( x ) α , (2)where α (cid:54) = 0 is a fixed infinitesimal number α ∈ ∗ R i [i]. Now, we apply the rulesof the function st ( · ) and exchange the application of F and st ( · ). Thus, wesearch for functions f with F ( st ( f (cid:48) ) , f, x ) = st ( F ( f (cid:48) , f, x )) = 0, i.e., we serachfor functions f with F ( f (cid:48) , f, x ) ∈ ∗ R i [i]. Example 1.
Let us take the differential equation F = Z − X . In standardand in non-standard analysis we would call f ( x ) = x a solution of this equation,3ecause 0 = dfdx ( x ) − x . In the given novel concept we first insert (2) to the Z -component of the polynomial F ( Z, Y, X ) and yield: F ( f (cid:48) ( x ) , f ( x ) , x ) = ( x + 3 x α + 3 xα + α ) − x α − x = α · (3 x + α ) . The polynomial f ( x ) = x is indeed a solution of the differential equation, butonly in the following sense: For all x ∈ ∗ R f [i] the expression F ( f (cid:48) ( x ) , f ( x ) , x ) isan infinitesimal number. In terms of real numbers, F can not be distinguishedfrom being zero, but in terms of hyperreal numbers, F is not zero (at least notfor all x ∈ ∗ R f [i], in this example only at x = − α ). The set T f ( F ) of zeros of F for a given f is very important in the next sections.Is this what we would call a solution? Yes, but... A heretical thought: Froman applied mathematical point of view, finding the solution s ( x ) of a differen-tial equation is like searching for an x -dependent recipe to approximate s ( x )numerically. More precisely, if we want to solve ordinary differential equations,we can do it, on the one hand, in an algebraic way. Then we restrict ourselvesto a purely symbolic description. Saying that, e.g., “exp( x ) is the solution ofa differential equation” does not provide a recipe to compute it. It has to betransformed into an algorithm for actually computing its values numerically.Only by this, we are able to leave the purely symbolic description and enterthe “real world” of applications. On the other hand, if we stay in a purelynumerical treatment of such an ordinary differential equation, then we do notactually solve it: In the deep core of our algorithms we only restrict our solu-tion procedures to some basic algebraic operations which would not allow for“really” computing expressions like exp( x ). Computing algebraic solutions ornumerically approximating the equation: We always seek for x -dependent lo-calized recipes to compute the searched function s . The only recipes that wecan perform are floating point operations on elements of algebraic fields. Wewill restrict ourselves here to polynomials f as the search space of our recipes.Our approach will, thus, be based on algebra and some concepts of finitism [9].Here is the new definition of what we will regard as a substitute for “solutionof a differential equation”. Definition 1
According to (2), a polynomial f ∈ C [ X ] is denoted as hyper-solution of F ∈ C [ Z, Y, X ] in D , if F ( f (cid:48) ( x ) , f ( x ) , x ) ∈ ∗ R i [i] for all x ∈ D . Example 2.
Let us consider the equation F = Z − Y with an initial valueof f (0) = 1, then in the set of polynomials f ∈ { C [ X ] , f (0) = 1 } a solution of F can not be found. The standard solution would be given by f ( x ) = exp( x ).Let us enter the “numerical way” to solve an equation like f (cid:48) = f . How dowe calculate exp( x ) in practice? The exponential exp( x ) is approximated by aTaylor polynomial, e.g. exp( x ) ≈ f ( x ) = 1 + x + x + x . If we insert the This also would be a hyper-solution in C , if we would allow for non-polynomials f . f into the differential equation, the resulting F -polynomial is F ( f (cid:48) ( x ) , f ( x ) , x ) = − x + 16 α + 12 αx + 12 α. The leading monomial − x remains. F has the form F = rx n + α G ( x, α )with n > r (cid:54) = 0. Surprisingly, this polynomial meets the requirementsfor a hyper-solution: F is infinitesimal for all x ∈ ∗ R i [i] = D . This is thespirit of Taylor polynomials, they “equal” the functions and their derivativesonly in “one point” - the center point. Any polynomial f which leads to an F -polynomial with vanishing constant term, is a hyper-solution of F in the setof infinitesimal numbers. For example, the same argumentation also holds forthe polynomial ¯ f = 1 + x + x + x which leads to an F -polynomial of theform: ¯ F = − x + 52 x + α + 3 αx + 12 α. ¯ F is infinitesimal for infinitesimal input values x . This polynomial has the form¯ F = P ( x ) + α G ( x, α ), where P does not have a constant term. Example 3.
The above considerations show, that F ( f (cid:48) , f, x ) in general hasmany different hyper-solutions f ( x ), if we restrict D to infinitesimal input values x , i.e., if we analyze F in the “monad” [3] of the center point 0. How can wefind these hyper-solutions? Taking the example F = ( f (cid:48) ) + f − f (0) = y at the center point 0. First, we fix the polynomial degree( d f = 2) and prepare the template polynomial f ( x ) = ax + bx + c. Then weinsert f into the differential equation and apply st ( · ) to F which provides st ( F ) = a x + 2 abx + (4 a + b + 2 ac ) x + (4 ab + 2 bc ) x + b + c − . The initial value condition provides the constant term of f which is c = y (inour case we chose y = 0). Now we proceed step by step through the monomials x k starting with k = 0: The constant term of st ( F ) should vanish, such that f is a hyper-solution of F for infinitesimal input values. Thus, the equation b + c − c = 0 this provides two possiblesolutions b = 1 or b = −
1. It is true, that the initial value problem leads to twodifferent standard solutions (sin( x ) and − sin( x )). We will proceed with b = 1.In principle we are already done: Any polynomial of the form f ( x ) = ax + x leads to a vanishing constant term in P , where F ( f (cid:48) , f, x ) = P ( x ) + α G ( x, α ).However, we can also take the higher monomials x k into account. For k = 1, theequation 4 ab + 2 bc = 0 with b = 1 and c = 0 has to be solved. This equationprovides a = 0. Now all coefficients are fixed a = 0 , b = 1 , c = 0. Interestingly,with these settings we get st ( F ) = P = x . Only one monomial remains, itturns out to be the leading one in F also regarding the infinitesimal terms. For the initial value condition f (0) = 1, this equation would not add any further infor-mation to the search of the coefficients. F ( Z, Y, X ) are polynomials only inmonomials
Z, XZ, Y, and in monomials of the form X n with a maximal degree m , see Tab.1. F F ( f (cid:48) , f, x ) one solution XZ + Z − x + 1) f (cid:48) − x + 1) Z − Y f (cid:48) − f exp( x )( X + 1) Z + 2 Y ( x + 1) f (cid:48) + 2 f ( x + 1) − Z − X − X f (cid:48) − x − x x + x Table 1: Some examples of differential equations which allow for hyper-solutions f such that F = rx n + α G ( x, α ). If n has to be higher than thegrade of G with regard to x (i.e. for α = 0), then f coincides with Taylorpolynomials of standard solutions.In these cases, the polynomial degree of F with regard to x (i.e. by setting α = 0) is the same as the polynomial degree d f of f (if this degree is at least m ). We apply the above method for finding a hyper-solution, such that F = rx n + α G ( x, α ), where n = d f and a given initial condition f (0) = y is valid.We get one condition for the coefficients from the initial value, f (0) = y . Weget further d f conditions for the coefficients, because all terms x n with n < d f have to vanish in F . In total, we get d f + 1 conditions for d f + 1 coefficients.The resulting polynomials f are Taylor approximations of the standard solutionof F . F F ( f (cid:48) , f, x ) one solution Z + Y − f (cid:48) ) + f − x )4( X + 1) Z + Y − x + 1)( f (cid:48) ) + f − √ x + 1)( X + 1) Z − Y ( x + 1) f (cid:48) − f exp( − ( x + 1) − )Table 2: Some examples of differential equations which allow for hyper-solutions f such that F = P ( x ) + α G ( x, α ).In those cases, when F ( Z, Y, X ) has terms like Z or like X Z , the poly-nomial degree of F in X is in general higher than the polynomial degree of f ,see Tab.2. If f is a Taylor approximation of the standard solution, it can notbe assured for such equations that P ( x ) is of the form rx n . However, it can beassured that the polynomial degree of P is higher than the polynomial degreeof G with regard to x . These considerations lead to the following definitions: Definition 2
A polynomial f ∈ C [ X ] is denoted as hyper Taylor approximationof F , if F ( f (cid:48) , f, x ) has the form F = rx n + α · G ( x, α ) with G ∈ C [ X, Y ] , n ∈ N > ,and r ∈ C , where the grade of G with regard to X is smaller than n . Definition 3
A polynomial f ∈ C [ X ] is denoted as hyper local approximationof F ∈ C [ Z, Y, X ] , if F ( f (cid:48) , f, x ) has the form F = P ( x ) + α · G ( x, α ) with G ∈ C [ X, Y ] , where the grade of G with regard to X is smaller than the gradeof P and where P does not have a constant term.
6s a summary of our findings we formulate the following
Theorem 1
Let F ∈ C [ Z, Y, X ] be a differential equation and f ∈ C [ X ] acomplex polynomial, then the following statements hold.(i) If and only if F ( f (cid:48) , f, x ) = α · G ( x, α ) with G ∈ C [ X, Y ] , then f is ahyper-solution of F in C .(ii) If f is a hyper Taylor approximation and r (cid:54) = 0 , then F ( f (cid:48) , f, x ) = 0 implies x ∈ ∗ R i [i] .(iii) If F ( f (cid:48) , f, x ) ∈ C [ x, α ] is lacking a constant term, then f is a hyper-solution of F in ∗ R i [i] .(iv) Every hyper-solution of F is also a hyper Taylor approximation. Everyhyper Taylor approximation is also a hyper local approximation. Only the second statement needs to be shown. (Case 1) Assume, that F = rx n + α · G ( x, α ) with r (cid:54) = 0. Furthermore, assume that x is a finite non-infinitesimal number. Then F is finite and non-infinitesimal. x is not a zero of F . (Case 2) Assume, that F = rx n + α · G ( x, α ) with r (cid:54) = 0. Furthermore, assumethat x (cid:54) = 0 such that we can divide by x n . The equation rx n + α · G ( x, α ) = 0is then equivalent to r + α · ˜ G (1 /x, α ) = 0, where ˜ G ∈ C [ X, Y ] is a suitablepolynomial. This equation can not be solved by an infinite number x , becausein this case 1 /x would be infinitesimal and st ( r + α · ˜ G (1 /x, α )) = r (cid:54) = 0. q.e.d. The rules for hyper local approximations and for the hyper Talyor approxima-tions in Sec. 2 are constructed in such a way, that f is related to the Taylorseries method for solving a differential equation at the center point x = 0.Whenever one wants to shift this center point to a different value x (cid:54) = 0, thenone has to analyze the differential equation F ( f (cid:48) ( x ) , f ( x ) , x + x ) instead. Example 4.
Take the equation F = xf (cid:48) −
1. Trying to find the hyper Taylorapproximation of this equation is like trying to expand ln( x ) at x = 0. If wewould like to extend the logarithm at x = 1, we have to solve the differentialequation F ( f (cid:48) , f, x + 1) which is equal to˜ F ( f (cid:48) , f, x ) = F ( f (cid:48) , f, x + 1) = ( x + 1) f (cid:48) − . (3)Indeed, the polynomial ˜ f ( x ) = x − x leads to ˜ F ( ˜ f (cid:48) , ˜ f , x ) = − x − α ( x + 1).Thus, ˜ f ( x ) is a hyper Taylor approximation of ˜ F . This procedure leads to thefollowing Definition 4
Let F ∈ C [ Z, Y, X ] define a differential equation F ( f (cid:48) , f, x ) , then F x ( f (cid:48) , f, x ) := F ( f (cid:48) , f, x + x ) is called the differential equation F at x . F = ( x + 1) f (cid:48) − F = xf (cid:48) − x = 1. At this point, f ( x ) = x − x is a hyperTaylor approximation of this equation .Can we transport a hyper Taylor approximation to a new center point? Thesituation is trivial, if the polynomial F ( Z, Y, X ) does not have an X -term, i.e.,if transportation of F to F x does not change the differential equation like for F = f (cid:48) − f or F = ( f (cid:48) ) + f −
1. In these cases, any suitable f for F is alsosuitable for F x . This leads to the following Definition 5
Let f ∈ C [ X ] be a hyper Taylor approximation of F . If f x ( x ) = f ( x ) is a hyper Taylor approximation of F x for all values x ∈ C , then thedifferential equation F is denoted as exponential-like solvable . We would expect that a transformation of the form f x ( x ) = f ( x + x ) isthe “correct” transformation rule . However, if a hyper Taylor approximation f leads to the polynomial F ( f (cid:48) , f, x ) = rx n + αG , then a substitution of x → x + x in F ( f (cid:48) , f, x ) turns a monomial rx n into a polynomial r ( x + x ) n . A hyperTaylor approximation turns into a hyper local approximation. Only in the caseof r = 0 this transformation is vaild. This means, only for hyper-solutionsthis transformation can be applied. This is the case, whenever the polynomial f solves F in standart analysis. The polynomial X which is a hyper Taylorapproximation of F = f (cid:48) − x , can thus be “transported” to the hyper Taylorapproximation X + 2 x X + x at the point x . More precise: Definition 6
Let f ∈ C [ X ] be a hyper Taylor approximation of F . If f x ( x ) = f ( x + x ) is a hyper Taylor approximation of F x for all values x ∈ C , thenthe differential equation F is denoted as polynomial-like solvable . What kind of transformation would “keep” the non-infinitesimal monomial rx n in F of a hyper Taylor approximation? A transformation of the type x → xx with x (cid:54) = 0 would do so. The equation F = xf (cid:48) −
1, is a corresponding example.It is not polynomial-like solvable, because a suitable polynomial f for F doesnot exist. It is also not exponential-like solvable. We will first introduce thedefinition and show its applicability afterwards. Definition 7
Let f ∈ C [ X ] be a hyper Taylor approximation of a differentialequation F ∈ C [ Z, Y, X ] at . If f x ( x ) = f ( xx ) is a hyper Taylor approxima-tion of F x for all values x (cid:54) = 0 ∈ C , then the differential equation F is denotedas logarithm-like solvable . Note, that indeed the polynomial X − X for X = ( x − x ) is the quadratic Taylorexpansion of ln( x ) at x = 1 In standard analysis: If a function f ( x ) (like exp( x )) solves F at 0, then f ( x + x ) solves F at x (like exp( x + x ) = exp( x ) · exp( x )). In our setting this is not true anymore. Thus,our approach is really different from standard analysis. roof. It will be shown that F = xf (cid:48) − f ∈ C [ X ] which is a hyper Taylor approximation of F =( x +1) f (cid:48) − f = f ( xx )is a hyper Taylor approximation of F x . F x ( ˜ f (cid:48) , ˜ f , x ) = ( x + x ) ˜ f (cid:48) −
1= ( x + x ) ˜ f ( x + α ) − ˜ f ( x ) α −
1= ( x + x ) f ( x + αx ) − f ( xx ) α −
1= ( x + x ) f ( xx + αx ) − f ( xx ) α −
1= ( x + x ) 1 x f ( xx + αx ) − f ( xx ) αx − (cid:0) x + x x (cid:1)(cid:0) f (cid:48) (cid:0) xx (cid:1) + R (cid:1) − (cid:0) xx + 1 (cid:1) f (cid:48) (cid:0) xx (cid:1) − R = ( z + 1) f (cid:48) ( z ) − R . Changing the infinitesimal quantity α to α/x changes the value of the derivativein (2). However, it only changes the value up to an infinitesimal difference R (the grades in x are not changed). Since R = R ( x + x ) x − and ( x + x ) x − is finite, R is also infinitesimal. Note, that ( z + 1) f (cid:48) ( z ) − R has thecorrect form for a hyper Taylor approximation, because f is a hyper Taylorapproximation of F , R is infinitesimal (in polynomial form), and the leadingmonomial is transformed from rx n into rx n x n by x → z . q.e.d. Example 5.
With a similar calculation one can show that F ( f (cid:48) , f, x ) = xf (cid:48) + n · f, (4)with a natural number n , is logarithm-like solvable . The standard analysissolution of this equation would be x − n . Example 6.
For the complicated example exp( − /x ) of standard analysis,where the Taylor series at x = 0 does not coincide with the function itself ,the differential equation is F ( f (cid:48) , f, x ) = 2 f − x f (cid:48) . (5)The equation (5) does not look like polynomial-like solvable. However, onewould have to check this for all hyper Taylor approximations. The polynomial For n = 1 a hyper Taylor approximation of F is given by f ( x ) = 1 − x + x . after continuation with exp( − / ) := 0 = 0 is a hyper-solution of this differential equation, i.e., a hyper Taylor approx-imation. The corresponding (transported) polynomial is f x ( x ) = f ( x + x ) = 0which also is a hyper Taylor approximation of the transported differential equa-tion. If this f = 0 is the only hyper Taylor approximation of (5), then it ispolynomial-like solvable.The three definitions of Sec. 3 provide possible recipes to “solve” ordinarydifferential equations. If we can find approximates at every center point x , i.e.,suitable polynomials f x for every F x , then we can (at least locally) “solve”ordinary differential equations, because we know how the set of “numericalrecipes” looks like locally for every number x (cid:54) = 0 ∈ C . How can we turnhyper-solutions in ∗ R i [i] into standard solutions of differential equations in C ? This section will follow the usual concept of numerical mathematics (like “walk-ing along the real axis” in Fig. 1). We construct solutions of the differentialequations with standard numerical tools [6] like step size control and like ad-justment of the polynomial degree. In all of the following cases, s ( t ) is not ahyper Taylor approximation of F , unless there exists a hyper-solution of F in C . It is just a good numerical approximation of a standard solution of F . Wewill present the change of paradigm in Sec. 5.Although the procedure in Sec. 3 provides hyper Taylor approximations ofa given differential equation F for all points x ∈ C , it does not seem to besatisfactory in terms of “solving the differential equation”. We would expectthat there is only one function for all points x ∈ C instead of a set of functions(approximates) for each point. How to glue these local approximations togetherto yield an approximate global solution? In numerical mathematics, differen-tial equations are treated in terms of initial value problems . In addition to F ( f (cid:48) , f, x ) = 0, we further define an initial condition f ( x ) = y to be satisfied.Solving an initial value problem like this in the context of this article, wouldmean to restrict the set of polynomials f to a certain subset f ∈ P ⊂ C [ X ],which meets the initial value condition P = { f ∈ C [ X ]; f ( x ) = y } . As anexample look at F = f (cid:48) − f . If we have found a polynomial which is a hyperTaylor approximation f of this equation, then every multiple λ · f also is a hy-per Taylor approximation in this special case. Only if we additionally ask for f (0) = 1, then solely polynomials with constant part 1 are valid. One exampleis f ( x ) = 1 + x + x ∈ P .The transportation mechanisms described in Sec. 3 not only have to trans-port the center point, but they also have to transport the initial value condition f ( x ) = y to the new center point t . Definition 8
If an initial value problem is given by a feasible set P of polyno-mials and by a differential equation F , then we call F t together with the feasibleset P t = { f ∈ C [ X ]; f ( X − t ) ∈ P } the initial value problem at t ∈ C . t ∈ C we“approximately solve” the initial value problem at t . Let f t ( x ) denote the“approximate solution” of the initial value problem at t . Then, one wouldexpect, that the function s ( t ) = f t (0) “approximately solves” the correspondinginitial value problem. Note, that f t ( x ) has the center point t and meets therequired (transported) initial value condition f ( x ) = y . Let us check thisna¨ıve way: Good Example.
This example shows how this transported solution of initialvalue problems works:Take the example of the initial value problem F = f (cid:48) − x , with f (0) = 0and the hyper-solution f ( x ) = x .We have to find a feasible hyper-solution for F and P . The equation is F = f (cid:48) − x − f with f ( −
1) = 0. Thepolynomial f ( x ) = ( x + 1) = x + 2 x + 1 is feasible, because f ∈ P ,and f is a hyper-solution of F .For the equation F = f (cid:48) − x −
4, we need a polynomial f with f ( −
2) =1. A corresponding hyper-solution is f ( x ) = ( x + 2) = x + 4 x + 4.Thus, f ( x ) = x is a hyper-solution at t = 0, f ( x ) = x + 2 x + 1 is ahyper-solution at t = 1, and f ( x ) = x + 4 x + 4 is a hyper-solution at t = 2.The three polynomials satisfy the (transported) initial value condition f (0) = 1.In this situation, we would expect, that a “solution” s ( t ) of the initial valueproblem is given by s (0) = f (0) = 0, and s (1) = f (0) = 1, and s (2) = f (0) = 4,which coincides with s ( t ) = t . Bad Example.
The next example shows, that this transportation mechanismis not valid for hyper Taylor approximations in general:Take the example of the initial value problem F = f (cid:48) − f , with f (0) = 1.A feasible hyper Taylor approximation is f ( x ) = 1 + x + x .We have to find a feasible hyper Taylor approximation for F and P . Theequation is the same F = f (cid:48) − f , but in this situation we search for apolynomial f with f ( −
1) = 1. The polynomial f ( x ) = 2 + 2 x + x is feasible, because f ∈ P , and f is a hyper Taylor approximation of F = F . Note, that f is just a multiple of f .For the equation F = f (cid:48) − f , we need a polynomial f with f ( −
2) = 1.A corresponding hyper Taylor approximation is f ( x ) = 1 + x + x .Thus, f ( x ) = 1 + x + x is a hyper Taylor approximation at t = 0, f ( x ) = 2 + 2 x + x is a hyper Taylor approximation at t = 1, and f ( x ) = 1 + x + x is a hyper Taylor approximation at t = 2.11he three polynomials satisfy the (transported) initial value condition f (0) = 1.In this situation, we would expect, that a “solution” s ( t ) of the initial value prob-lem is given by s (0) = f (0) = 1, and s (1) = f (0) = 2, and s (2) = f (0) = 1,which does not coincide with exp( t ). This is a bad approximation, because f t only hyper-solves the differential equation locally and the initial value condition(at a different position x (cid:54) = t ) is out of this infinitesimal range. For higherorder polynomials, like f = (cid:80) n =1 x n n ! , this procedure provides better estimates of exp( t ). Asking for polynomials f with “infinte” grade to solve the initialvalue problem (including problems of convergence of Taylor series) is not thespirit of this article.Second idea (controlling the grade of the polynomials): For a numericaltreatment of initial value problems we could further restrict the set of polyno-mials to “better fits”, if we do not want to deal with “infinite grades”. A hyperTaylor approximation f t ∈ P t of an initial value problem { F, P } at t is a goodapproximation, if e.g. | st ( F t ( f (cid:48) , f, x )) | < (cid:15), for all | x + 12 t | ≤ | t | . (6)Such a condition assures, that f t is a good numerical approximation of the dif-ferential equation in the whole “interval” [ − t,
0] and that f t has the correctinitial value. This additional condition (6) further restricts the set of possiblepolynomials f t . It’s like a discretization-based adjustment of the grade of thepolynomial. Again s ( t ) = f t (0) is the resulting numerical solution.Third idea (step size control): An alternative approach using a step sizecontrol on t is possible, too. Let us start with a differential equation F and aninitial condition P . A hyper Taylor approximation f of this initial value problemis also a good numerical solution of the differential equation in a certain “region”,such that there is e.g. a ∆ > | st ( F ( f (cid:48) , f, x )) | ≤ (cid:15) for all | x | ≤ ∆. Wecan evaluate f at any point t ∈ C with | t | ≤ ∆ for a good approximation of the“solution”. Select one value t and f ( t ) = y . Then we proceed with the sameargumentation for the next step by replacing the differential equation F with F t and the initial value condition P with the new condition f (0) = y . In eachstep k of this procedure we get a step size t k and a hyper Taylor approximation f ( k ) . The numerical solution is given by s ( (cid:80) nk =1 t k ) = f ( n ) (0). Numerical Example.
As an example take the initial value problem F =( x + 1) · f (cid:48) − f (0) = 0 = s (0).(1) A hyper Taylor approximation is given by f (1) ( x ) = x − x . We select astep size: t = . This leads to f (1) ( ) = . For f = (cid:80) n =1 x n n ! the estimates are s (1) = and s (2) = which is close to thecorresponding values of exp( t ). Also an empty set can be the result of this restriction. F = ( x + ) · f (cid:48) − f (2) (0) = . A hyper Taylor approximation is given by f (2) = + x − x which has been found by the logarithm-like solution procedure. Againselecting t = . This yields f (2) ( ) ≈ . s (0) = 0, s (0 .
5) = 0 . s (1) ≈ . , ln(1 + 0 . ≈ . , ln(1 + 1) ≈ . insola with α ∗ = 0 .
001 and a maximal polynomial degree of n = 40. Top: The roots T ∗ ⊂ C for the initial value problem F = f (cid:48) − f and f (0) = 1. Bottom Left: The roots T ∗ ⊂ C for the initial value problem F = ( x + 1) f (cid:48) − f (0) = 0. Bottomright: Again, the same roots like in the left figure, but for every point t ∗ ∈ C the absolute value of s ( t ∗ ) is shown on the z -axis. The problem of numerical methods described in the last section is the following:We want to get away from the center point x = 0. Once we leave this point, theselected polynomials only approximate the solution of the equation. Thus, weshould stay at the center point or in its “monad”, see Fig. 1. Hyper local approx-13mations can not be distinguished from a solution of the differential equation inthis case. Through the glasses of real numbers, moving away by infinitesimalsteps is not different from looking at x = 0. Moreover, an infinitesimal valuefor F is also like F = 0 in real numbers. If we insert one hyper-solution f in D = ∗ R i [i], then the corresponding polynomial F ( f (cid:48) , f, x ) only has a finite setof infinitesimal roots denoted as T f ( F ). A root β ∈ T f ( F ) provides a class ofsequences, which exactly solves the equation for the given polynomial f and thegiven class of sequences α for computing f (cid:48) . This means, that the solution s should be constructed in the following way: Given a hyper-solution f of F in D = ∗ R i [i], we compute the infinitesimal roots of F ( f (cid:48) , f, · ) leading to the set T f ( F ). For every β ∈ T f ( F ) we define s ( β ) = f ( β ), because it is f ( β ) whichis inserted into F to provide a zero of F . We repeat this procedure for everyhyper-solution f . Here it has to be said that only a relation s : ∗ R i [i] ↔ ∗ R i [i](not necessarily a function) is constructed in this way, because the sets of rootscan intersect for different polynomials f . The interesting point is, that by com-puting the roots of F for every given hyper-solution f , we determine at whichpoints β the algebraic recipe f is a valid solution method. This procedure is verydifferent from trying to find one solution function f for which F is infinitesimalon C . Now we have recipes in the ring ∗ R i [i]. How can we make these recipesvisible in C ? Numerical experiments.
The described relation s is numerically not ac-cessible, because we apply infinitesimal numbers, which are not represented innumerical routines. However, if we want to find an approximate representationof this relation with non-infinitesimal numbers, then we maybe simply replace α with a very small real number α ∗ in the above considerations. Here comesthe infinitesimal solution algorithm ( insola ). Repeat for all grades n :1. We first compute a hyper Taylor approximation f (with r (cid:54) = 0) of theinitial value problem for a fixed polynomial grade n .2. Then we determine F ( f (cid:48) ( x ) , f ( x ) , x ) ∈ ∗ R f [i][ x ] using (2). F has rootsonly in the set of infinitesimal numbers.3. In the expression for F we replace α with a small real value α → α ∗ andyield F ∗ ∈ C [ x ].4. Then, we compute all roots of F ∗ , which are finite complex numbers t ∗ ∈ T ∗ f .5. For every number t ∗ ∈ T ∗ f we plot the relation ( t ∗ , f ( t ∗ )) ∈ s .A MATLAB TM -code that can be used to visualize and do experiments withthe different initial value problems is in the Appendix [4]. In this code wereplaced the search for hyper Taylor approximations by computing the Taylorpolynomials (of the known solutions) directly. The roots found by the algorithmfor two different initial value problems are shown in Fig. 2 (on the top exp( x ) andin the bottom row ln( x + 1)). We yield the same picture like on the bottom left14igure 3: These figures show results that have been found by insola with α ∗ = 0 .
001 and a maximal polynomial degree of n = 40 with hyper local ap-proximations instead of hyper Taylor approximations. Top Left: The roots T ∗ ⊂ C for the initial value problem F = ( f (cid:48) ) + f − f (0) = 0. Topright: Restriction to the real valued roots (on the x-axis) and the expected realvalued results of sin( x ) (red curve). Bottom Left: After eliminating the non-infinitesimal roots. Bottom right: Comparing the real-valued results from insola with sin( x ) after eliminating the non-infinitesimal roots.15igure 4: The insola algorithm also finds the expected solution exp( − x +1) )of F = 2 f − ( x + 1) f (cid:48) after elimination of the non-infinitesimal roots. Left:The complex valued roots. Right: The roots versus the absolute value of theproposed solution s on the z-axis (blue dots). The values coincide with theexpected function values (red circles).(however with different function values for the points), if we solve the differentialequation F = xf (cid:48) + 2 f which is also logarithm-like solvable according to (4).In order to compare these representations with the expected function valuesin C , the insola points are plotted as blue points, whereas, the results fromstandard analysis are plotted as red circles (bottom right, Fig. 2). The resultsfrom insola coincide with our expectations about the standard solutions of thecorresponding differential equations. Hyper local approximations.
In order to test the algorithm insola for thecase of hyper local approximations, we let it run for F = ( f (cid:48) ) + f − f (0) = 0 and the additional condition that the coefficientof x in f is positive. We expect the solution sin( x ). In the top right plot inFig. 3, the real valued roots x ∈ T ∗ f are plotted versus the actual value of theapproximated solution s ( x ) = f ( x ) with blue crosses. However, these crossesdo not coincide with the red curve, which shows the expected values ( x versussin( x )). Here another reason shows up, why insola is based on hyper Taylorapproximations. Hyper local approximations can lead to finite non-infinitesimalroots of F . The polynomial f , however, is only valid hyper-solution within therange of infinitesimal numbers. This means, the finite non-infinitesimal valuesare out of the region where F is hyper-solved by f . If we insert a finite real value α ∗ into F in step (3.) of insola for numerical reasons, then the set T ∗ f whichhas to be constructed in the fourth step will always consist of finite complexnumbers. We can not distinguish between numbers that stem from “turning α into a real number” or from the finite non-infinitesimal roots of F . If wecould distinguish these two cases, then we could sort out the non-infinitesimalroots. The algorithm that we propose to sort out non-infinitesimal roots from T ∗ f ( F ) uses the assumption that F is of the form F ( x, α ) = P ( x ) + αG ( x, α ),16igure 5: These plots show different representations (i.e. different values for α ∗ , the symbol x means α ∗ = 0 . . . x )-function. Left: Changes of thecomplex roots. The smaller α ∗ , the more the roots converge to zero. Right:Changes of the real-valued roots and comparison with the expected sin-functionvalues.with a polynomial P . The non-infinitesimal roots of T f ( F ) are assumed to beclose to the roots of P , which can be accessed by setting α → F ( x, α ). Weapplied this method to the example of Fig. 3 and indeed end up with roots andapproximated values, which coincide with sin( x ) (in the real valued roots and-not shown- also in the complex roots). It also works for other examples, seeFig.4. The strategy is always the same: In the set of infinitesimal numbers, thehyper local approximations are the substitute for a “solution” of F . Now, weonly regard the infinitesimal roots of F . By the replacement of α with α ∗ wecan represent or approximate the scaled-up roots of F in the complex plane. Inthis way, this procedure turns out to provide an approximate solution also inthe set of complex numbers. Note that this “scaling-up” trick also works fordifferent choices of α ∗ , see Fig. 5 Usually solving a differential equation means to search for a function f : C → C such that F ( f (cid:48) ( x ) , f ( x ) , x ) is an infinitesimal number for all x ∈ C . If wereduce (change) the input domain of f to infinitesimal numbers, then finding f is not a problem anymore. There are many possible polynomials f whichare hyper-solutions of F in this domain. In this new approach, the choice of f defines the set T f ( F ) of points at which F = 0. By this, we define our solution s . The interesting observation is, that it is possible to scale-up this relation s into the set of complex numbers providing good approximations to the standardsolutions of F . Maybe this new paradigm (“Serach for x -dependent polynomialsto compute s in ∗ R i [i]”) allows for a different perspective onto the existence anduniqueness (is s a function?) of the solution of differential equations, in general.17 ppendix: Matlab TM Code syms xsyms asyms fsyms fder %derivative of f% standard solution for comparison with s() in the plotsfs=log(x+1); %only leading monomial in P%fs=exp(x); %only leading monomial in P%fs=sin(x); %polynomial P%fs=sin(sqrt(x+1)); %polynomial P%fs=(x+1)^(-2); %only leading monomial in P%fs=x^3+x^2; %polynomial P=0%fs=exp(-(x+1)^(-2)); %polynomial P% define small alpha and maximal grade of the polynomialsalpha=0.001;grades=40;figure(1);hold on;figure(2);hold on;for i=2:grades% Taylor polynomialsf=taylor(fs,i);% differential equation (select the correct one)dgl=(x+1)*fder-1; %for log(x+1)%dgl=fder-f; %for exp(x)%dgl=fder^2+f^2-1; %for sin(x)%dgl=4*(x+1)*fder^2+f^2-1; %for sin(sqrt(x+1))%dgl=(x+1)*fder+2*f; %for (x+1)^(-2)%dgl=fder-3*x^2-2*x; %for x^3+x^2%dgl=(x+1)^3*fder-2*f; %for e^(-(x+1)^(-2))%prepare F(f’,f,x) =expand(subs(dgl, fder, (subs(f,x, x+a)-f)/a));%numerical approximation: insert a finite alphaFn=expand(subs(F,a,alpha));%roots of the polynomial FnC=sym2poly(Fn);Tf=roots(C);Tf=unique(Tf);%insert a=0 to identify non-infinitesimal rootsF0=expand(subs(F,a,0));%roots of the polynomial F0 (to exclude from Tf)C0=sym2poly(F0);Tf0=roots(C0);Tf0=unique(Tf0);%excluding non-infinitesimal rootsfor j=1:length(Tf0)if(Tf0(j)~=0)[val, ind]=sort(abs(Tf-Tf0(j)));Tf=Tf(ind(2:end));endendfor j=1:length(Tf)% for the roots that are real plot s as a "graph"if(isreal(Tf(j)))figure(1);plot(Tf(j), subs(f,x,Tf(j)),’.b’);plot(Tf(j), subs(fs,x,Tf(j)),’ro’);end% for complex valued roots r plot Re(r), Im(r) against the% absolute value of s(r)figure(2);plot3(real(Tf(j)),imag(Tf(j)), abs(subs(f,x,Tf(j))),’.b’);plot3(real(Tf(j)),imag(Tf(j)), abs(subs(fs,x,Tf(j))),’ro’);endend eferences [1] Pringsheim A. and Faber G. 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