Novel exact solutions, bifurcation of nonlinear and supernonlinear traveling waves for M-fractional generalized reaction Duffing model and the density dependent M-fractional diffusion reaction equation

Diverse Precise Traveling Wave Solutions Possessing Beta Derivative of the Fractional Differential Equations Arising in Mathematical Physics

“…We have also depicted some of the obtained solutions graphically (3D surface graphs and 2D line plots) and concluded that the results we obtained are accurate, efficient, and versatile in mathematical physics. It is worth to noticing that compared to previous works [25,26,44,45]; the results obtained in this paper are presented for the first time. Lastly, it can be concluded that our offered methods are more effective, reliable, and powerful, which give bounteous consistent solutions to NLPFDEs arise in different fields of nonlinear sciences.…”

“…Recently, many powerful methods for obtaining exact solutions of nonlinear partial differential equations (NLPDEs) have been presented, such as exponential rational function method [21], exp a function, and the hyperbolic function methods [22]. ðG ′ /GÞexpansion method [23,24], ðG′/G, 1/GÞ-expansion method [25,26], Sardar-subequation method [27], new subequation method [28], Riccati equation method [29], homotopy perturbation method [30], extended direct algebraic method [31], Kudryashov method [32], Exp-function method [33], the modified extended exp-function method [34], F-expansion method [35], the Backlund transformation method [36], the extended tanh-method [37], Jacobi elliptic function expansion methods [38], extended sinh-Gordon equation expansion method [39], and different other methods [40][41][42][43].…”

Diverse Exact Soliton Solutions of the Time Fractional Clannish Random Walker’s Parabolic Equation via Dual Novel Techniques

“…There are several definitions of fractional derivatives such as Riemann Liouville [2], conformable fractional derivative [3], beta derivative [4], and new truncated M-fractional derivative [5] are available in literature. Many powerful methods for obtaining exact solutions of nonlinear fractional PDEs have been presented as Hirota's bilinear method [6], sinecosine method [7], tanh-function method [8], exponential rational function method [9], Kudryashov method [10], sine-Gordon expansion method [11], modified ðG ′ /GÞ -expansion method [12], extended ðG ′ /GÞ-expansion method [13], ðG ′ /GÞ-expansion method [14], tanh-coth expansion method [15], Jacobi elliptic function expansion method [16], first integral method [17], sardar-subequation method [18], new subequation method [19], extended direct algebraic method [20], exp ð−ϕðηÞÞ method [21], Exp a function method [22], ð1/G′Þ, ðG′/G, 1/GÞ, and modified ðG′/ G 2 Þ − expansion methods [23,24], Kudryashov method [25], modified expansion function method [26], new auxiliary equation method [27], extended Jacobi's elliptic expansion function method [28], extended sinh-Gordon equation expansion method [29], modified simplest equation method [30], and many more.…”