Hypercomplex Systems with Some Applications of White Noise Analysis

Applied functional analysis has many applications in other branches of mathematics, such as differential equations, numerical analysis, stochastic calculus, calculus of variations, quantum field theory, etc. Among of these applications, we interest in stochastic differential equations (SDEs). There were many authors studied this subject. Recently, Okb El Bab, Zabel, Bin Dehaish, Ghany, Hyder and Zakarya studied some important subjects related to the connection between the Harmonic analysis in hypercomplex systems (HCSs) and Gaussian calculu. So the main objective of this thesis is to use a round applied functional analysis. We gave a constructions of non-Gaussian white noise analysis (WNA) using the theory of HCSs. We introduced an elements of non-Gaussian Wick calculus based on HCSs. We applied this construction to find non-Gaussian white noise functional solutions (WNFSs) of stochastic partial differential equations (SPDEs). The KdV equation is one of the essential nonlinear equations in mathematical physics. So, we can get non-Gaussian (WNFSs) of stochastic KdV and coupled KdV equations.

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Dr. M. Zakarya. He is Lecturer in Department of Mathematics, Al-Azher University, Assiut, Egypt. He is a member of the Egyptian mathematical society, a member of the Egyptian syndicate of scientific professions (2006-now). He has published research articles in reputed international journals of mathematical and mathematical physics.