Diversity of exact solutions to the conformable space–time fractional MEW equation

Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

“…More interestingly, the fractional one-soliton in Figure 2 has a bend in the middle; such phenomena often occur on the edges of flat beaches. In addition to the same amplitude k 2 1 /2 and the similar bell profile known from Equation (23), to gain more insights in the dynamics of the one-solitons in Figure 1 mathematically, we insert a = 0, b = 2, c = 3, k 1 = 1, p 1 = 1/2, σ = 1, and ξ ð0Þ 1 = 0 into Equation ( 24); then, the velocity along the x -axis of the one-solitons can be derived by the expression…”

“…Some novel spatial structures obtained in fractional KP Equation (3) cannot be derived from corresponding integral-order KP Equation (2). This is due to the irreplaceable role of fractional calculus [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Different soliton solutions to the modified equal-width wave equation with Beta-time fractional derivative via two different methods

“…Consider the modified equal width wave (MEW) equation [10] with beta-time fractional derivative given as…”

“…This equation has been solved by different analytical methods such as: the tanh −function method [8,9], the ansatz and improved (G /G)−expansion methods [10]. But the extended Jacobi elliptic expansion function method and Kudryashov method have not been exercised for the above mentioned model with a fractional beta derivative operator.…”

Closed-form wave structures of the space-time fractional Hirota–Satsuma coupled KdV equation with nonlinear physical phenomena

“…For all the attention, it is an indispensable task to attain the closed-form wave structures of the fractional differential equations (FDEs). Various meaningful and truthful approaches have been interpolated for achieving the closed-form wave structures of FDEs, including the -expansion approach [4,5], extended Jacobi elliptic function expansion method [6], improved sub-equation scheme [7], modified fractional reduced differential transform method [8], sub-equation method [9], singular manifold method [10], fractional homotopy method [11], fractional reduced differential transform method [12], modified ( ′/ ) G G -expansion approach [13], extended modified mapping method [14], Sine-Gordon expansion method [15], extended trial equation method [16], iterative method [17], simplest equation method [18], ansatz scheme [19], F-expansion method [20], modified Kudryashov method [21], extended mapping method [22], homo separation analysis method [23], modified simple equation method [24], reduced differential transform scheme [25], modified extended mapping method [26], functional variable method [27], extended direct algebraic method [28], Darcy's law rule [29], function transformation method [30], the variation of ( ′/ ) G G -expansion method [31], differential transform method [32], unified scheme, ( ′/ ) G G -expansion approach [33,34], Kudryashov method [35], new extended Kudryashov process [36], Sine-cosine approach [37], auxiliary equation scheme [38] and many other techniques…”