Analytical Methods to Investigate Exact Solutions for Space-Time Fractional Differential Equations Arising in the Real Physical Phenomena of Mathematical Physics and Biology

Abstract

Fractional derivatives are most important to accurate nonlinear modeling of various real-world difficulties in applied nonlinear science and engineering incidents especially in the fields of crystal, optics and quantum mechanics even in biological phenomena. The investigation of exact solutions of such nonlinear models has great important to visualize the nonlinear dynamics. We consider the space-time fractional nonlinear differential equations for pulse narrowing transmission lines model, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM), complex Schrodinger and biological population models, the complex time fractional Schrodinger equation (FSE) and low-pass electrical transmission lines equation (ETLE) are studied with the effective unified method, Jacobi elliptic expansion function integral technique, generalized Kudryshov technique, modified simple equation (MSE) method respectively. As a result, we get some solitary wave solutions in the form of hyperbolic and combo hyperbolic-trigonometric :functions including both stable and unstable cases. We obtain kink wave, bright bell wave, dark bell wave, combo periodic-rogue waves, combo M-W shaped periodic-rogue waves in stable cases, and singular kink type in unstable solitonic natures. Lastly, we proposed an Improved Kudryashov method for solving any nonlinear fractional differential models. We apply the proposed approach to the nonlinear space­time fractional model leading wave spread in electrical transmission lines (s-tfETL), the space­time M-fractional Schrodinger-Hirota (s-tM-fSH) and the time fractional complex Schrodinger (tfcS) models to verify the effectiveness of the propose approach. The implementations of the introduced new technique on the models provide us periodic envelope, exponentially changeable soliton envelope, rational, combo periodic-soliton and combo rational-soliton solutions, which are much interesting phenomena in the nonlinear sciences. Beside the scientific derivation of the analytical findings, we represent the results graphically for clear visualization of the dynamical properties.