Colombeau Algebra Research Articles

On a fractional Cauchy problem with singular initial data

Abstract This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: D α x ( t ) = F ( t , x ( t ) ) x ( 0 ) = x 0 , \left\{\begin{array}{l}{D}^{\alpha }x\left(t)=F\left(t,x\left(t))\hspace{1.0em}\\ x\left(0)={x}_{0},\hspace{1.0em}\end{array}\right. where x 0 {x}_{0} is the singular generalized function and F satisfies L ∞ {L}^{\infty } logarithmic type, D α {D}^{\alpha } is the Caputo derivative of order m − 1 < α < m m-1\lt \alpha \lt m with m ∈ N * m\in {{\mathbb{N}}}^{* } , which we will confirm to be present in Colombeau algebra. The Gronwall lemma is used in Colombeau’s algebra to establish the main results. To illustrate our theoretical analysis, we ended our work with an example.

Differential geometry and general relativity with algebraifolds

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra A for which the module of derivations of A is finitely generated projective. Over R as the base ring, this class of algebras includes the algebra C∞(M) of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion M→N makes C∞(M) into an algebraifold with C∞(N) as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.

Products of distributions in Colombeau algebra

In this paper, we evaluate some products of distributions in Colombeau algebra of generalized functions. In the classical theory of Schwartz distributions, multiplication of distributions is not defined for two arbitrary singular distributions. The properties of the Colombeau algebra allow us to calculate products of singular distributions which are not defined in the classical theory. The notion of association in Colombeau algebra of generalized functions allows us the results obtained in this way to be considered as products in the classical theory of distributions. The definition of the Colombeau product of distributions can be considered as generalization of their classical product in Schwartz theory.

Generalized solutions for time $\psi$-fractional evolution equations

This paper focuses on the time fractional wave problem with the use of a new fractional derivative in Colombeau algebra. Using Banach's fixed point theorem and Laplace transforms, we give and prove the integral solution of the problem. In Colombeau's algebra, The existence and uniqueness of the solution are demonstrated using the Gronwall lemma.

Generalized solutions of the Cauchy problem involving $\Phi$-Caputo fractional derivatives

The main objective of this research paper is to embed $\Phi$-Caputo fractional derivative in the Colombeau algebra of generalized functions and we investigate the existence and uniqueness of the Cauchy problem involving $\Phi$-Caputo fractional derivatives in the extended Colombeau algebras. Finally, we give anexample of how the ideas presented in the document can be applied.

Generalized solution of burger equation

In this paper, The existence and uniqueness of the generalized solution of the Burger equation is studied with initial conditions are distributions (elements of Colombeau algebra). then we look at the association notion in conjunction with the classic solution.

Generalised solutions to linear and non-linear Schrödinger-type equations with point defect: Colombeau and non-Colombeau regimes

For a semi-linear Schrödinger equation of Hartree type in three spatial dimensions, various approximations of singular, point-like perturbations are considered, in the form of potentials of very small range and very large magnitude, obeying different scaling limits. The corresponding nets of approximate solutions represent actual generalised solutions for the singular-perturbed Schrödinger equation. The behaviour of such nets is investigated, comparing the distinct scaling regimes that yield, respectively, the Hartree equation with point interaction Hamiltonian vs the ordinary Hartree equation with the free Laplacian. In the second case, the distinguished regime admitting a generalised solution in the Colombeau algebra is studied, and for such a solution compatibility with the classical Hartree equation is established, in the sense of the Colombeau generalised solution theory.

Solving the Fractional Schrödinger Equation with Singular Initial Data in the Extended Colombeau Algebra of Generalized Functions

This manuscript aims to highlight the existence and uniqueness results for the following Schrödinger problem in the extended Colombeau algebra of generalized functions. 1 / ı ∂ / ∂ t u t , x − △ u t , x + v x u t , x = 0 , t ∈ R + , x ∈ R n , v x = δ x , u 0 , x = δ x , where δ is the Dirac distribution. The proofs of our main results are based on the Gronwall inequality and regularization method. We conclude our article by establishing the association concept of solutions.

On the Study of Singular Fractional Evolution Problems via a Generalized Fixed Point Theorem in Colombeau Algebra

On the Study of Singular Fractional Evolution Problems via a Generalized Fixed Point Theorem in Colombeau Algebra

Existence and Uniqueness of Generalized Solution of Sine-Gordon Equation by Using Fixed Point in Colombeau Algebra

Existence and Uniqueness of Generalized Solution of Sine-Gordon Equation by Using Fixed Point in Colombeau Algebra

On the existence of almost automorphic generalized solutions to some generalized differential equations

This paper is devoted to study some regularity of almost automorphic and asymptotic almost automorphic generalized solution of the differential equation d dtu(t) = Au(t) + f(t), in the framework of the Colombeau algebras. Under certain assumptions about the second member we showed that the generalized solution is an asmptotically almost automorphic in the sense of genaralized functions.

Generalized Solution of Transport Equation

This paper proves the existence and uniqueness of solution of irregular transport problem with variable speed and initial data in the Colombeau algebra G, and some important proprieties of Colombeau algebra. The existence of distribution solutions to some classes of such equations is proven.

Regularity of nonlinear generalized functions: A counterexample in the nonstandard setting

Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of G∞-regularity, which reduces to C∞-regularity when restricted to Schwartz distributions. Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of G∞-regularity does not hold.

The point free Colombeau geometry in singular general relativity

The problem statement. We argue that the canonical interpretation of the Schwarzschild spacetime in contemporary general relativity is wrong and that revision is needed. And we argue that the Schwarzschild solution is impossible to treat classically, since the Levi-Cività connection is not available for the whole Schwarzschild spacetime (Sch,gijSch (t r, , ,θϕ)) ; where Sch=×(({r ≥ 2m} {∪ ≤ ≤0 r 2m})×S2) ; but it can only be treated by using an embedding of the classical Schwarzschild metric tensor gijSch; ,i j =1,2,3,4 into Colombeau algebra δ(4,Σ),Σ= ={r 2m} {∪ =r 0} supergeneralized functions. The classical Schwarzschild spacetime could be extended up to the distributional semi-Riemannian manifold endowed on the tangent bundle with the Colombeau distributional metric tensor. The aim. The development of new physical interpretation for the distributional curvature scalar (Rε( )r )ε and square scalar (Rεμν( )r Rμνε, ( )r )ε, (Rερσμν( )r Rρδμνε, ( )r )εis aimed. Results. The Schwarzschild solution using Colombeau distributional geometry without leaving Schwarzschild coordinates (t r, , ,θϕ) is studied. We obtain that the distributional Ricci tensor and the curvature scalar are δ-type, (R rε( ))ε=−m rδ( − 2m) ,>0 . The practical value. As distributional square scalars are essentially nonclassical Colombeau type distributions: (Rεμν( )r Rμνε, ( )r )ε, (Rερσμν( )r Rρσμνε, ( )r )ε∈(3 )\ ′(3 ), this provides a new physical interpretation for the distributional curvature scalar (R rε( ))ε and square scalars (Rεμν( )r Rμνε, ( )r )ε, (Rερσμν( )r Rρδμνε, ( )r )ε.

Nonlinear generalized functions on manifolds.

In this work, we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators. This produces a generalization of previous theories in a form which is suitable for applications to differential geometry. The generalized Lie derivative is introduced and shown to extend the Lie derivative of Schwartz distributions. A new feature of this theory is the ability to define a covariant derivative of generalized scalar fields which extends the covariant derivative of distributions at the level of association. We end by sketching some applications of the theory. This work also lays the foundations for a nonlinear theory of distributional geometry that is developed in a subsequent paper that is based on Colombeau algebras of tensor distributions on manifolds.

Maximal closed ideals of the Colombeau Algebra of Generalized functions

In this paper we investigate the structure of the set of maximal ideals of {{mathcal {G}}}(Omega ). The method of investigation passes through the use of the m-reduction and the ideas are analoguous to those in Gillman and Jerison (Rings of Continuous Functions, N.J. Van Nostrand, Princeton, 1960) for the investigation of maximal ideals of continuous functions on a Hausdorff space K.

Regular generalized solutions to semilinear wave equations

The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In classical terms, the result says that the semilinear wave equations under consideration have global classical solutions up to a rapidly vanishing error.

Propagation of singularities for generalized solutions to nonlinear wave equations

The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof is based on a fixed point theorem in a suitable ultrametric topology on the subset of Colombeau solutions possessing the required regularity. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equations, and shows that the same behavior is attained in the case of non-classical, generalized solutions.

Generalized solution to multidimensional cubic Schrödinger equation with delta potential

This article addresses the Cauchy problem for the defocusing cubic Schrodinger equation in 2D and 3D and the equation with a delta well potential in 3D. Solutions belong to the Colombeau algebra of generalized functions $$\mathcal {G}_{C^1,H^2}$$ (see Nedeljkov et al. in Proc R Soc Edinb Sect A Math 135:863–886, 2005). The physically significant homogeneous problem in 2D and 3D has not yet been treated in this framework, whereas no classical results exist on the equation with delta potential. The paper contains the construction of unique generalized solutions for both of these problems. One could also find an assertion about compatibility with classical solutions for 2D and 3D equations without delta potential.

Full and Special Colombeau Algebras

AbstractWe introduce full diffeomorphism-invariant Colombeau algebras with addedε-dependence in the basic space. This unites the full and special settings of the theory into one single framework. Using locality conditions we find the appropriate definition of point values in full Colombeau algebras and show that special generalized points suffice to characterize elements of full Colombeau algebras. Moreover, we specify sufficient conditions for the sheaf property to hold and give a definition of the sharp topology in this framework.