2006•Integral Transforms and Special FunctionsRequires access

Heaviside generalized functions and shock waves for a Burger kind equation

Francisco Villarreal

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Abstract

The purpose of the work is to study the existence and nonexistence of shock wave solutions for the Burger equations. The study is developed in the context of Colombeau's theory of generalized functions (GFs). This study uses the equality in the strict sense and the weak equality of GFs. The shock wave solutions are given in terms of GFs that have the Heaviside function, in x and (x, t) variables, as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function, in ℝ n and ℝ n ×ℝ, in the distributional limit sense.

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What this paper is about

The purpose of the work is to study the existence and nonexistence of shock wave solutions for the Burger equations. The study is developed in the context of Colombeau's theory of generalized functions (GFs). This study uses the equality in the strict sense and the weak equality of GFs. The shock wave solutions are given in terms of GFs that have the Heaviside function, in x and (x, t) variables, as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function, in ℝ n and ℝ n ×ℝ, in the distributional limit sense.

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Available abstract

The purpose of the work is to study the existence and nonexistence of shock wave solutions for the Burger equations. The study is developed in the context of Colombeau's theory of generalized functions (GFs). This study uses the equality in the strict sense and the weak equality of GFs. The shock wave solutions are given in terms of GFs that have the Heaviside function, in x and (x, t) variables, as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function, in ℝ n and ℝ n ×ℝ, in the distributional limit sense.

Key concepts: Heaviside step function, Mathematics, Generalized function, Context (archaeology), Shock wave, Mathematical analysis, Function (biology), Shock (circulatory)

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