Differential algebraic interpretation with dense singularities of the order completion method

Introduction

The generalized algebraic functions with the help of sequential approach has been discussed in this paper. We will choose elements ranges 0 to 1. Let us consider occurs that conserves derivatives of distributions.It can be easily said that the impression of distributions as well as generalized functions with nowhere dense singularities are little bit hard to make it compatible. Thus, it's our interest to explore that one can achieve such kind of embedding fordistributions into a different manner for algebra of generalized functions, without modification in the above described property related to the existence of outcomes of nonlinear partial differential equations. Moreover, we demonstrate that this construction is said to be compatible with Colombeau's. In this regard, we can further classified an ideal analogous to Colombeau's, in a way to embed

which is denoted as a sub-algebra (we can also say that it is used to preserve the product of   -functions). The amalgamation of the constructed algebras & distributions which contain good embedding-properties should definitely diminishes the false suggestion of non-compatibility between distributions as well as generalized functions with dense singularities.

Algebraic Properties & Definitions

Here we find a new algebra instead of Rosinger's formulations . We can also say that if and only if for all "a" belongs to " O there is neighborhood V of a as well as 0  such that 1 ) (    over the given neighborhood for 0    .

Rosinger's Algebra & Embedding Distributions in

. Also assume that "a" belongs to " O , there occurs an open neighborhood V of "a" in such a manner for small ε,

Theorem 1:There occurs a natural surjective morphism of differential algebras from

Now, in case of a nowhere dense as well as closed set, we can see that for unit nets

, this is the only reason that the definition does not depend on the chosen unit net. Furthermore we can say that it is independent of the representatives:

Again for any specified unit net

. This represents that  is surjective. Moreover, it is obviously a linear morphism. It can be concluded that the product of two unit nets is nothing but a unit net, it can also belongs to a morphism of algebras. At the end, we can also check that  commutes with partial derivatives.

In case of Rosinger's algebra

, generally convolution method is used to embed . First of all we prove another embedding result.

Theorem 2: There occurs a natural embedding of differential algebras from

Proof:As earlier we described that

 is nothing but differential algebra with the point wise definitions of dot as well plus sign (+, '.') and i  .

 so this map is absolutely defined. Thus, it can be easily demonstrate a morphism of differential algebras. However, some embedding findings

.Now reveal some definitions and say a sequence

is bounded without depends on ε.

A linear map with relation

is a linear embedding of (ii) Let us say S be a smooth part map. The map

Is said to be linear embedding of

A Colombeau Algebra with Non-Smooth Representatives& its PDEs solution

In order to provide a sense to the fundamental results in

is well defined and therefore it can be described as

)) ,....., ( (: ) ,......., ( :

Lemma 2:The definition of  S put more emphasis on a well-defined map

Proof: First of all we have to find that it does not depends on the representatives, or we can say, that

. This can be follows from the characterization discussed in literature combined with the trivial fact that, if for some fixed values of ε and 'a' belongs to " O ,

 

. It can also be defined as

Further, we can describe point wise values of elements belongs to

. However, we can illustrate the value of for all 'a' belongs to " O by the following relation:

The above relation described that this limit exists for some

. If it occurs, it is clearly mentioned that this does not depends on the choice of  , since for

is dense in " O , so it can be determines the limit Now we focus on the analytic Cauchy data described by the following equation

(2) Thus, we will illustrate an algebra with same kind of

Conclusion

The proposed work given an idea that the choice of . In this paper we identified smooth nonstandard maps which differs only on the point infinitely close to dense close subset. The solution of analytic partial differential equation has been discussed.

References

1. Kovalevskaia S. ZurTheorie der partiellendifferentialgleichung, Journal fur die reine und angewandte Mathematik, 1875; 80:1-32.
2. Colombeau JF. New Generalized Functions and Multiplication of Distributions, vol. 84, North-Holland, Amsterdam, The Netherlands, 1984.
3. Rosinger EE. Global version of the Cauchy-Kovalevskaia theorem for nonlinear PDEs, Acta Applicandae Mathematicae, 1990; 21(3):331-343.
4. Oberguggenberger MB, Rosinger EE. Solution of Continuous Nonlinear PDEs through Order Completion, North Holland, Amsterdam, The Netherlands, 1994, 181.
5. Evans LC. Partial Differential Equations, of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1998, 19
6. Baire R, Lecons Sur Les Fonctions Discontinues, Collection Borel, Paris, France, 1905.
7. Arnold VI. Lectures on Partial Differential Equations, Springer, Berlin, Germany, 2004.
8. van der Walt JH. The order completion method for systems of nonlinear PDEs revisited, ActaApplicandaeMathematicae, 2009; 106(2):149-176,
9. Beattie R, Butzmann HP. Convergence Structures and Applications to Functional Analysis, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.
10. Anguelov R. Dedekind order completion of C(X) by Hausdorff continuous functions. Quaest. Math. 2004; 27:153-170.
11. Anguelov R, Rosinger EE. Solving large classes of nonlinear systems of PDE's. Comput. Math. Appl. 2007; 53:491-507.
12. Anguelov R, van der Walt JH. Order convergence structure on C (X). Quaest. Math. 2005; 28:425-457.
13. Egorov YV. A contribution to the theory of generalized functions, Russian Math Surveys 45: 1-49 (translated from Russian), 1990
14. Lindstr€om T. An invitation to Nonstandard Analysis. In: Cutland N (ed) Nonstandard Analysis and its Applications, Cambridge: Univer Press, 1988, 1-105.
15. Oberguggenberger M. Multiplication of Distributions and Applications to Partial Differential Equations, Essex: Longman, 1992
16. Oberguggenberger M, Todorov T. An Embedding of Schwartz Distributions in the Algebra of Asymptotic Functions, Internat J Math & Math Sci. 1998; 21:417-428.
17. Robinson A. Non-Standard Analysis, Amsterdam: North-Holland, 1966.
18. Rosinger EE. Nonlinear Partial Differential Equations: Sequential and Weak Solutions. Amsterdam: North-Holland, 1980.
19. Rosinger EE. Generalized Solutions of Nonlinear Partial Differential Equations. Amsterdam: North-Holland, 1987.
20. Rosinger EE. Nonlinear Partial Differential Equations, An Algebraic View of Generalized Solutions, Amsterdam: NorthHolland, 1990.