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This paper presents a fractional approach to study the mathematical model of tsunami wave propagation along a coastline of an ocean. Fractional Reduced Differential Transform Method (FRDTM) is used to analyze this model. The present model has been studied on the shallow-water assumption. It is represented by a time-fractional coupled system of non-linear partial differential equations. Solutions to the proposed model for different coastal slopes and ocean depths have been obtained. Effects of coast slope and sea depth variations on tsunami wave velocity and wave amplification have been demonstrated at different time levels and different orders α . The obtained results are compared with Elzaki Adomian Decomposition Method (EADM) to validate the proposed method for an order α = 1 .

In the current study, we establish a fractional reduced differential transform method, which is successfully applied to obtain the analytical approximate solutions of the nonlinear generalized fractional-order Fitzhugh–Nagumo equation. Also, we considered three different cases of generalized fractional-order Fitzhugh–Nagumo equation for different values of $$\alpha $$ . The fractional derivative is considered in the context of the Caputo derivative. The obtained results show that the proposed technique is efficient, and convenient to implement fractional-order differential equations. We compared the approximate solutions and the exact solutions of the partial fractional differential equations for integer order. The approximate solutions are rapidly convergences of the exact solutions and also, this method reduced numerical calculation.

In this paper, the counter – current imbibition phenomena in a heterogeneous porous media is studied with the consideration of two types of porous materials like volcanic and fine sand and Adomian decomposition method is applied to find the saturation of wetting phase and the recovery rate of the reservoir. A simulation result is developed here to study the effect of heterogeneity, capillarity and relative permeability on saturation rate and to obtain an optimum recovery rate of the reservoir with the choices of some interesting parametric value.

The concept of calculus to fractional order is widely used to analyze the complex non-linear phenomena depend on the time. In the study of unsaturated ow in soils, Brook-Corey’s model (BCM) is the most applicable model. In the present work, we considere Richards’ equation employed by BCM. We propose a fractional model involving Caputo’s sense derivative. Fractional order Richards’ equation is analyzed using FRDTM. For the validity of the method, the obtained solutions are compared with exact solutions for order $$\alpha = 1$$ . Also the comparison between errors obtained by exact solutions and FRDTM solutions and errors obtained by exact solutions and q - HATM solutions shows that the proposed method is very effective.

The objective of this paper is to present a reliable approach to compute an approximate analytical solution of magneto-hydrodynamic (MHD) Jeffery–Hamel flow by using a new modification of Adomian decomposition method. The approximate solution of this problem is calculated here in the form of a rapidly convergent series at one and both grid points. Results for velocity profiles in divergent and convergent channels are presented for different values of Hartmann and Reynolds numbers. The validity and applicability of this new technique are illustrated through a comparison between reproducing Kernel Hilbert Space Method (RKHSM) (Inc et al., 2013), Modified Homotopy Perturbation Method (MHPM) (Singh, 2014) and numerical solutions. Keywords: Jeffery–Hamel flow, Modified Adomian Decomposition Method (MADM), Nonlinear differential equation

In this paper, the counter-current imbibition phenomena with gravity, inclination effect and with Corey’s model in heterogeneous porous media is discussed and Adomian decomposition method is applied to find the saturation rate of wetting phase and the recovery rate for both medium. Finally a simulation result is developed to study the saturation and recovery rate of the oil reservoir with dimensionless time for some interesting choices of parametric data with gravity and inclination effect. This problem has a great importance in oil recovery process.

In this paper, the counter-current imbibition phenomenon in two phase fluid through fracture porous media is discussed and Adomian decomposition method is applied to find the saturation of wetting phase and the recovery rate of the reservoir. A simulation result is developed for the saturation of wetting phase in fracture matrix and in porous matrix for some interesting choices of parametric value to study the recovery rate of the oil reservoir with dimensionless time. This problem has a great importance in the oil recovery process.

In this paper, the counter - current imbibition phenomenon is discussed in an inclined heterogeneous porous media with the consideration of two types of porous materials like volcanic sand and fine sand. Adomian decomposition method is applied to find the saturation of wetting phase and the recovery rate of the reservoir. Finally, a simulation result is developed to study the saturation of wetting phase and the optimum recovery rate of reservoir with the choices of some interesting parametric values. This problem has a great importance in the field of oil recovery process.