Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation

New Numerical Results for the Time-Fractional Phi-Four Equation Using a Novel Analytical Approach

“…Recently, many new fractional-order operator types have been introduced to the literature by some scholars, and established its fundamental properties [1][2][3][4][5][6]. Fractional theory is widely used to explain many properties of phenomena such as nanotechnology [7], optics [8], human diseases [9], chaos theory [10], and others . As an example, the tumor-immune surveillance model has been investigated in [33].…”

Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

“…In fact, fractional calculus provides the mathematical modeling of some important phenomena like social and natural in a more powerful way than the classical calculus. During the last few decades, many applications were reported in many branches of science and engineering such as chaotic systems [6,7], fluid mechanics [8], viscoelasticity [9], optimal control problems [10,11], chemical kinetics [12,13], electrochemistry [14], biology [15], physics [16], bioengineering [17], finance [18], social sciences [19], economics [20,21], optics [22], chemical reactions [23], rheology [24], and so on. Due to the importance of FDEs, the solutions of them are attracting widespread interest.…”

Complex and Real Optical Soliton Properties of the Paraxial Non-linear Schrödinger Equation in Kerr Media With M-Fractional

“…In the research papers, researchers have been noted several computational methods for solving NPDEs, building separate solitons, and other alternatives for distinct types of NPDEs such as, the Haar wavelet method [1], the homotopy perturbation method [2], the Adomian decomposition method [3,4], the shooting method [5][6][7][8], the sine-Gordon expansion method [9][10][11][12], the inverse scattering method [13], the sinh-Gordon expansion method [14][15][16], the tan(φ (ξ ) /2)-expansion method [17,18], the inverse mapping method [19], modified exp (−ϕ (ξ ))-expansion function method [20][21][22][23], the decomposition-Sumudu-like-integral-transform method [24], a functional variable method [25], the Bernoulli sub-equation function method [26][27][28], modified exponential function method [29], the modified auxiliary expansion method [30], the Riccati-Bernoulli sub-ODE method [31], the extended trial equation method [32,33], and tanh function method [34,35]. Also, different methods have been used to solve fractional differential equation such as, the finite difference method [36], the improved Adams-Bashforth algorithm [37,38], Adams-Bashforth-Moulton method [39], the extended fractional sinh-Gordon expansion method …”