Fractional calculus is a name for the theory of integrals and derivatives of arbitrary (real or complex) order, which unify and generalize the notions of integer-order differentiation and n-fold integration. In recent years, the study of fractional differential equations (FDEs) has gained considerable popularity and importance due to the accurate description of widespread applications in various areas of science and engineering like mechanics, physics, visco-elasticity, viscoelasticity, fluid dynamics, hydrology, chemical kinetics, biological models, image processing, mathematical finance, and so on. The main advantage of fractional-order (non-integer order) differential operator over integer-order differential operators is their non-local nature. They depend on all the past values of the function, which help them describe memory to a complex system. Due to power-law memory, the fractional-order derivatives do not satisfy the standard form of the Leibniz rule, chain rule, and semigroup property. These properties of fractional derivatives are called unusual properties, which are essential characteristic properties because they help us to describe memory. So, the derivation of exact solutions of FDEs is challenging for the reasons mentioned. It is well-known that the exact solutions of FDEs play a significant role in understanding the behaviours of complex systems and materials in science and engineering. Due to the unusual properties of fractional-order derivatives, no analytic method exists to solve the fractional-order nonlinear ODEs and PDEs. In recent years, mathematicians and engineers have concentrated on developing numerical and analytical methods for finding solutions to FDEs, such as Lie symmetry analysis, iteration methods, homotopy analysis, and invariant subspace methods. Recent investigations show that the invariant subspace method is an algorithmic and powerful analytical tool for deriving exact solutions for nonlinear FPDEs.
Here, I would like to point out that to the best of my knowledge, no one has developed the invariant subspace method for higher-dimensional nonlinear fractional PDEs. So, the aim of this proposed project is to develop and generalize the invariant subspace method for solving nonlinear higher-dimensional fractional PDEs. More precisely, this study focuses on how to derive various finite separable exact solutions for the fractional nonlinear convection-reaction-diffusion equation along with initial and boundary conditions using the invariant subspace method. Additionally, note that the fractional PDEs provide a tool to study both parabolic and hyperbolic types of PDEs simultaneously, which studies the intermediate processes between diffusion and wave behaviours. In addition, another aim of this project is to explain how to develop the invariant subspace method for finding finite separable exact solutions of initial value and boundary value problems for multi-component coupled systems of nonlinear fractional PDEs in higher dimensions. In particular, the applicability and effectiveness of the method have been demonstrated for finding the exact solutions of multidimensional nonlinear fractional convection-diffusion-reaction-wave equations along with initial and boundary conditions. This systematical development of the method will help to understand or solve the large class of fractional nonlinear PDEs which arise in science and engineering. So, I assure you that the proposed research work is new and essential for understanding the behaviours of complex systems and materials in science and engineering.
Department of Mathematics, School of Physical Sciences, Coimbatore
CSIR-UGC- NET or GATE or Good Percentage in B.Sc., and M.Sc.(Mathematics/Applied Mathematics)
Assistant Professor
Department of Mathematics
School of Engineering, Coimbatore