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Studies an analytic solution and a reliable numerical approximation of linear and non‐linear wave equations by using the Adomian decomposition method. The solution is calculated in the form of a series with easily computable components. The non‐homogeneous problem is quickly solved by observing the self‐cancelling “noise terms” where the sum of the components vanishes in the limit. Several linear or non‐linear partial differential equations are considered and their numerical approximate solutions are compared with its numerical analytic solutions by applying the Adomian decomposition method and using a computer algebraic system (MATLAB). The numerical results show the effectiveness of the method for these types of equations.

The main purpose of this work is the development of a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. These types of equations describe a variety of physical models in the vibrations of structures, nonlinear optics, quantum field theory and solid-state physics, etc.

Dirichlet boundary conditions cannot be handled easily by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and a numerical algorithm is developed. The proposed algorithm reduced the hyperbolic-type wave equations into a system of first-order ordinary differential equations (ODEs) in time variable. Then, stability-preserving SSP-RK54 scheme and the Thomas algorithm are used to solve the obtained system. The stability of the algorithm is also discussed.

A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from the schemes developed (Abbas et al., 2014; Nazir et al., 2016) and depicts the computational modelling of hyperbolic-type wave equations.

To the best of the authors’ knowledge, this technique is novel for solving hyperbolic-type wave equations and the developed algorithm is free from quasi-linearization process and finite difference operators for time derivatives. This algorithm gives better results than the results discussed in literature (Dehghan and Shokri, 2008; Batiha et al., 2007; Mittal and Bhatia, 2013; Jiwari, 2015).

This paper aims to develop a new 3-level implicit numerical method of order 2 in time and 4 in space based on half-step cubic polynomial approximations for the solution of 1D quasi-linear hyperbolic partial differential equations. The method is derived directly from the consistency condition of spline function which is fourth-order accurate. The method is directly applied to hyperbolic equations, irrespective of coordinate system, and fourth-order nonlinear hyperbolic equation, which is main advantage of the work.

In this method, three grid points for the unknown function w(x,t) and two half-step points for the known variable x in spatial direction are used. The methodology followed in this paper is construction of a cubic spline polynomial and using its continuity properties to obtain fourth-order consistency condition. The proposed method, when applied to a linear equation is shown to be unconditionally stable. The technique is extended to solve system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the method.

The paper provides a fourth-order numerical scheme obtained directly from fourth-order consistency condition. In earlier methods, consistency conditions were only second-order accurate. This brings an edge over other past methods. In addition, the method is directly applicable to physical problems involving singular coefficients. Therefore, no modification in the method is required at singular points. This saves CPU time, as well as computational costs.

There are no limitations. Obtaining a fourth-order method directly from consistency condition is a new work. In addition, being an implicit method, this method is unconditionally stable for a linear test equation.

Physical problems with singular and nonsingular coefficients are directly solved by this method.

The paper develops a new fourth-order implicit method which is original and has substantial value because many benchmark problems of physical significance are solved in this method.

This study aims to analyse the non-linear losses of a porous media (stack) composed by parallel plates and inserted in a resonator tube in oscillatory flows by proposing numerical correlations between pressure gradient and velocity.

The numerical correlations origin from computational fluid dynamics simulations, conducted at the microscopic scale, in which three fluid channels representing the porous media are taken into account. More specifically, for a specific frequency and stack porosity, the oscillating pressure input is varied, and the velocity and the pressure-drop are post-processed in the frequency domain (Fast Fourier Transform analysis).

It emerges that the viscous component of pressure drop follows a quadratic trend with respect to velocity inside the stack, while the inertial component is linear also at high-velocity regimes. Furthermore, the non-linear coefficient b of the correlation ax + bx² (related to the Forchheimer coefficient) is discovered to be dependent on frequency. The largest value of the b is found at low frequencies as the fluid particle displacement is comparable to the stack length. Furthermore, the lower the porosity the higher the Forchheimer term because the velocity gradients at the stack geometrical discontinuities are more pronounced.

The main novelty of this work is that, for the first time, non-linear losses of a parallel plate stack are investigated from a macroscopic point of view and summarised into a non-linear correlation, similar to the steady-state and well-known Darcy–Forchheimer law. The main difference is that it considers the frequency dependence of both Darcy and Forchheimer terms. The results can be used to enhance the analysis and design of thermoacoustic devices, which use the kind of stacks studied in the present work.

The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM).

Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method.

No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler.

A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.

We use the Adomian decomposition method to study a non‐linear diffusion‐convection problem (NDCP). The decomposition method has been applied recently to a wide class of non‐linear stochastic and deterministic operator equations involving algebraic, differential, integro‐differential and partial differential equations and systems. The method provides a solution without linearization, perturbation, or unjustified assumptions. An analytic solution of NDCP in the form of a series with easily computable components using the decomposition method will be determined. The non‐homogeneous equation is effectively solved by employing the phenomena of the self‐cancelling ‘noise terms’ whose sum vanishes in the limit. Comparing the methodology with some known techniques shows that the present approach is highly accurate.

The purpose of this paper is to study the effect of injection and suction on velocity profile, skin friction and pressure distribution of a Casson fluid flow between two parallel infinite rectangular plates approaching or receding from each other with suction or injection at the porous plates.

The governing Navier–Stokes equations are reduced to the fourth-order non-linear ordinary differential equation through the similarity transformations. The approximated analytic solution based on the Homotopy perturbation method is given and also compared with the classical finite difference method.

From this study, the authors observed that the skin friction is less in non-Newtonian fluids compared to Newtonian fluids. The use of non-Newtonian fluids reduces the pressure in all the cases compared to Newtonian and hence load-carrying capacity will be more. As γ value increases velocity, skin friction and pressure decreases. When γ is fixed, it is observed that skin friction and pressure is minimum for A=0.5 and maximum when A=−0.5. The result of this study also shows that the effect of suction on the velocity profiles, pressure and skin friction is opposite to the effect of injection.

The present work analyzes the characteristic of non-Newtonian fluid having practical and industrial applications.

This paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.

HPM was used in this study.

Solution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.

Coupled non-linear shallow water wave equations are solved.

This paper studies the stability properties of a class of nonlinear output‐feedback control system. By using a control transform and constructing Liapunov function, the sufficient conditions of the asymptotic stability for the nonlinear control system are presented. The result in this paper includes some existing results.