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Multiplication for solutions of the equation grad f = M grad g
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2009 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, Vol. 59, no 10, 1412-1430 p.Article in journal (Refereed) Published
Abstract [en]

Linear first-order systems of partial differential equations (PDEs) of the form ∇ f = M ∇ g, where M is a constant matrix, are studied on vector spaces over the fields of real and complex numbers. The Cauchy-Riemann equations belong to this class. We introduce on the solution space a bilinear *-multiplication, playing the role of a nonlinear superposition principle, that allows for algebraic construction of new solutions from known solutions. The gradient equation ∇ f = M ∇ g is a simple special case of a large class of systems of PDEs, admitting a *-multiplication of solutions. We prove that any gradient equation has the exceptional property that the general analytic solution can be expressed as *-power series of certain simple solutions.

Place, publisher, year, edition, pages
2009. Vol. 59, no 10, 1412-1430 p.
Keyword [en]
Cauchy-Riemann equations; Multiplication of solutions; Overdetermined systems of PDEs; Power series
National Category
URN: urn:nbn:se:liu:diva-21201DOI: 10.1016/j.geomphys.2009.07.006OAI: diva2:240897
Available from: 2009-09-30 Created: 2009-09-30 Last updated: 2009-09-30

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Jonasson, Jens
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Applied MathematicsThe Institute of Technology

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