This article presents exact and approximate solutions of the seventh order time-fractional Lax's Korteweg-de Vries (7TfLKdV) and Sawada-Kotera (7TfSK) equations using the modification of the homotopy analysis method called the q-homotopy analysis method. Using this method, we construct the solutions to these problems in the form of recurrence relations and present the graphical representation to verify all obtained results in each case for different values of fractional order. Error analysis is also illustrated in the present investigation. Keywords Lax's seventh-order Korteweg-de Vries equation • Sawada-Kotera seventh-order equation • q-Homotopy analysis method • Fractional derivative Mathematics Subject Classification 26A33 • 34A12 • 35R11 • 35Q53 Consider the seventh-order time-fractional Lax's Korteweg-de Vries (7TfLKdV) and Sawada-Kotera (7TfSK) equations of the form D α

This paper presents analytical‐approximate solutions of the time‐fractional Cahn‐Hilliard (TFCH) equations of fourth and sixth order using the new iterative method (NIM) and q‐homotopy analysis method (q‐HAM). We obtained convergent series solutions using these two iterative methods. The simplicity and accuracy of these methods in solving strongly nonlinear fractional differential equations is displayed through the examples provided. In the case where exact solution exists, error estimates are also investigated.

In this present investigation, we proposed a reliable and new algorithm for solving time‐fractional differential models arising from physics and engineering. This algorithm employs the Shehu transform method, and then nonlinearity term is decomposed. We apply the algorithm to solve many models of practical importance and the outcomes show that the method is efficient, precise, and easy to use. Closed form solutions are obtained in many cases, and exact solutions are obtained in some special cases. Furthermore, solution profiles are presented to show the behavior of the obtained results in other to better understand the effect of the fractional order.

In this paper, we present analytical-approximate solution to the time-fractional nonlinear coupled Jaulent-Miodek system of equations which comes with an energy-dependent Schrödinger potential by means of a residual power series method (RSPM) and a q-homotopy analysis method (q-HAM). These methods produce convergent series solutions with easily computable components. Using a specific example, a comparison analysis is done between these methods and the exact solution. The numerical results show that present methods are competitive, powerful, reliable, and easy to implement for strongly nonlinear fractional differential equations.

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