Research of Jean-Daniel Djida

My research line is strongly connected to my goal in building an enduring team in Nonlinear Analysis of PDEs, Shape Optimization, Optimal Control, Deep Neural Networks with their Applications, to speed up the launched dynamics of conducting high-quality research in Mathematical subjects directly related to the sustainable development of Africa and Cameroon in particular. To achieve these objectives in collaboration with the AIMS-Cameroon Research Center and other University partners and collaborators, detailed plans for research and networking activities are scheduled. For links to my publications, see Publications.

Analysis of Optimal Control and Shape Optimization

Geometric variational problems arise in various applications e.g. in physics, engineering, and biology. Classical examples are the shape of solid bodies minimizing air resistance, area minimizing surfaces with prescribed boundary properties, or the shape of membranes giving rise to the lowest fundamental mode. In current mathematical language, these and other geometric variational problems can be formulated as optimization problems for domain-dependent functionals on given classes of subdomains of Rn or of Riemannian, Hyperbolic manifolds with or without boundary. They are usually called shape optimization problems and most generally, we are interested in the geometrical analysis of optimal shapes problems that involve other types of questionings such as symmetry, convexity, overdetermined problems, isoperimetric inequalities, etc. Our attention is drawn by these main questions driven by local or nonlocal operators.

Fractional Calculus and its Applications in Networks and Deep Neural Networks

Fractional calculus, the field of Mathematics dealing with operators of differentiation and integration of arbitrary real or even complex order, extends many of the modeling capabilities of conventional calculus and integer-order differential equations and finds its application in various scientific areas, such as physics, mechanics, engineering, economy, finance, biology, chemistry, Networks, etc.

Apart from fractional Calculus as its own subject, I am also interested in both direct and inverse problems for fractional differential equations of different orders (sub-parabolic, pseudo-parabolic, parabolic-hyperbolic, etc.).

I am pro-active in working on the development of the fractional calculus and my work is influenced by the requirements coming from applications but also includes extensive studies of certain aspects of the theory.

Functional Analysis and Analysis of PDEs

Important questions in connection with abstract linear and nonlinear differential equations on Banach spaces concern the wellposedness of these equations (existence, uniqueness and continuous dependence on the data) and the qualitative and quantitative behavior of solutions. The latter include regularity, positivity, approximation properties and asymptotic behavior such as stability, periodicity, instability. With methods of functional analysis (Banach space theory and operator theory), harmonic analysis and function theory, conclusions can be drawn about diffusion phenomena, wave phenomena and transport phenomena, among other things.

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