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In this manuscript, a new hybrid technique viz Sawi transform-based homotopy perturbation method is implemented to solve one-dimensional shallow water wave equations. In general, the quantities involved with such equations are commonly assumed to be crisp, but the parameters involved in the actual scenario may be imprecise/uncertain. Therefore, fuzzy uncertainty is introduced as an initial condition. The main focus of this study is to find the approximate solution of one-dimensional shallow water wave equations with crisp, as well as fuzzy, uncertain initial conditions. First, by taking the initial condition as crisp, the approximate series solutions are obtained. Then these solutions are compared graphically with existing solutions, showing the reliability of the present method. Further, by considering uncertain initial conditions in terms of Gaussian fuzzy number, the governing equation leads to fuzzy shallow water wave equations. Finally, the solutions obtained by the proposed method are presented in the form of Gaussian fuzzy number plots. [ABSTRACT FROM AUTHOR]

In this paper, the fractional reduced differential transform method (FRDTM) is used to obtain the series solution of time-fractional seventh-order Sawada–Kotera (SSK) and Lax's KdV (LKdV) equations under initial conditions (ICs). Here, the fractional derivatives are considered in the Caputo sense. The results obtained are contrasted with other previous techniques for a specific case, α = 1 revealing that the presented solutions agree with the existing solutions. Further, convergence analysis of the present results with an increasing number of terms of the solution and absolute error has also been studied. The behavior of the FRDTM solution and the effects on different values α are illustrated graphically. Also, CPU-time taken to obtain the solutions of the title problems using FRDTM has been demonstrated. [ABSTRACT FROM AUTHOR]

A widescale description of various phenomena in science and engineering such as physics, chemical, acoustics, control theory, finance, economics, mechanical engineering, civil engineering, and social sciences is well described by nonlinear fractional differential equations (NLFDEs). In turbulence, fluid dynamics, and nonlinear biological systems, applications of NLFDEs can also be found. NLFDEs are believed to be powerful tools to describe real‐world problems more precisely than the differential equation of the integer‐order. In this research, we have used the fractional reduced differential transform method (FRDTM) to find the solution of the time‐fractional Sawada‐Kotera‐Ito seventh‐order equation. The novelty of the FRDTM is that it does not require any discretization, transformation, perturbation, or any restrictive conditions. In addition, compared to other methods, this approach needs less calculation. For special cases of an integer and noninteger orders, computed results are compared with existing results. Present results are in good agreement with the existing solutions. Here, the fractional derivatives are considered in the Caputo sense. Convergence analysis of the results has also been studied with the increasing number of terms of the solution. [ABSTRACT FROM AUTHOR]

Purpose This paper aims to present solutions of uncertain linear and non-linear shallow water wave equations. The uncertainty has been taken as interval and one-dimensional interval shallow water wave equations have been solved by homotopy perturbation method (HPM). In this study, basin depth and initial conditions have been taken as interval and the single parametric concept has been used to handle the interval uncertainty.Design/methodology/approach HPM has been used to solve interval shallow water wave equation with the help of single parametric concept.Findings Previously, few authors found solution of shallow water wave equations with crisp basin depth and initial conditions. But, in actual sense, the basin depth, as well as initial conditions, may not be found in crisp form. As such, here these are considered as uncertain in term of intervals. Hence, interval linear and non-linear shallow water wave equations are solved in this study using single parametric concept-based HPM.Originality/value As mentioned above, uncertainty is must in the above-titled problems due to the various parametrics involved in the governing differential equations. These uncertain parametric values may be considered as interval. To the best of the authors’ knowledge, no work has been reported on the solution of uncertain shallow water wave equations. But when the interval uncertainty is involved in the above differential equation, then direct methods are not available. Accordingly, single parametric concept-based HPM has been applied in this study to handle the said problems. [ABSTRACT FROM AUTHOR]

The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa–Holm (CH), modified Camassa–Holm (mCH), and Degasperis–Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform method (FRDTM). The main benefits of using this technique are that linearization is not required for this method and therefore it reduces complex numerical computations significantly compared to the other existing methods such as the perturbation technique, differential transform method (DTM), and Adomian decomposition method (ADM). Small size computations over other techniques are the main advantages of the proposed method. Obtained results are compared with the solutions carried out by other technique which demonstrates that the proposed method is easy to implement and takes small size computation compared to other numerical techniques while dealing with complex physical problems of fractional order arising in science and engineering. [ABSTRACT FROM AUTHOR]