Numerous Novel Analytical Techniques for Solving Nonlinear Partial Differential Equations Arising in Physical Systems

Abstract

Partial differential equations (PDEs) are widely used to model a wide range of physical phenomena and are of crucial significance in numerous scientific and engineering fields. In recent years, fractional-order differential equations, being powerful generalizations of ordinary and PDEs to non-integer orders, have received significant attention for their remarkable effectiveness in modelling diverse real-world phenomena in numerous scientific and engineering disciplines. The fractional partial differential equation has been occurring gradationally in several study fields over the past two decades. Numerous critical phenomena in bioengineering, viscoelasticity, cosmology, biomathematics, signal analysis, traffic flow, biomedicine, electrochemistry, financial systems, and many others may be better understood using this fractional differential equation. The fractional calculus is a non-integer order extension of classical differentiation and integration. The tools and approaches of fractional calculus are utilized in almost every modern science and engineering field. One of the foremost significant nonlinear evolution equations in nonlinear optics is the nonlinear Schrödinger equation (NLSE), and numerous exact soliton solutions can be obtained by using various approaches. The NLSE is a nonlinear partial differential equation that is found in several chemistry, physics, and engineering fields. A significant aspect of this NLSE in a nonlinear dispersive medium is that it interprets wave propagation. The NLSE is widely used in a variety of domains, including plasma physics, semiconductor materials, fluids, and many more. Obtaining the exact solution to nonlinear equations is one of the most challenging tasks in engineering, applied mathematics, and physics. Exact solutions consistently provide an excellent description of the behavior of the phenomenon being investigated. Although, for some equations, obtaining these solutions is extremely challenging and, in some instances, impossible. Consequently, numerous analytical methods have been suggested for finding the exact solutions to nonlinear problems. By employing numerous strategies, various types of exact soliton and solitary wave solutions for a variety of nonlinear problems have been obtained in this study, such as kink singular, singular, bright, dark, combined dark-bright, the double periodic singular, the periodic singular, the dark singular, the dark kink singular, the periodic solitary singular, breather, W -shaped, bell-shaped, anti-kink, and kink-shaped solitons. In order to illustrate the physical significance of the acquired solutions and by giving particular values to undefined parameters, the graphical representation of certain derived solutions has been shown.