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This paper aims to present a numerical solution of non‐linear Burger's equation using differential quadrature method based on sinc functions.

Sinc Differential Quadrature Method is used for space discretization and four stage Runge‐Kutta algorithm is used for time discretization. A rate of convergency analysis is also performed for shock‐like solution. Numerical stability analysis is performed.

Sinc Differential Quadrature Method generates more accurate solutions of Burgers' equation when compared with the other methods.

This combination, Sinc Differential Quadrature and Runge‐Kutta of order four, has not been used to obtain numerical solutions of Burgers' equation.

The purpose of this paper is to discuss a numerical method for solving Fredholm integral equations of the first kind with degenerate kernels and convergence of this numerical method.

By using sinc collocation method in strip, the authors try to estimate a numerical solution for this kind of integral equation.

Some numerical results support the accuracy and efficiency of the stated method.

The paper presents a method for solving first kind integral equations which are ill‐posed.

The purpose of this paper is to build a phase-type risk model with stochastic return on investment and random observation periods to characterize the ruin quantities under which the insurance company may take effective investment strategies to avoid bankruptcy.

By the Markov property and Ito’s formula, this paper derives the integro-differential equations in which the interclaim times follow a phase-type distribution. Using the sinc method, this paper obtains the approximate solutions of the expected discounted penalty function. The numerical examples are given to verify the robustness of the proposed sinc method.

This paper discloses the relationship between the investment strategy and initial surplus level. The insurance company with a high initial surplus level prefers high risk portfolios to earn more profit. Contrarily, the insurance company would invest low risk portfolios to avoid bankruptcy. In addition, this paper shows that a short observation period would bring higher ruin probability.

The risk model is distinct in that a phase-type risk model is constructed with stochastic return on investment and random observation periods. These considerations in the risk model are in sharp contrast to the setting in which the stochastic return on investment is observed continuously. In practice, the insurance company only can periodically observe the surplus level to check the balance of the book. This setting, therefore, is difficult to adopt. This paper develops a sinc method to solve the approximate solutions of the expected discounted penalty function.

The purpose of this paper is to derive a dynamic equation for modelling the behaviour of smectic‐C liquid crystals under the effect of an electric field.

The model equation is solved using a finite difference approximation, method of lines and pseudo‐spectral methods. The solutions are compared for accuracy and efficiency. Comparison is made of the efficiency of finite differences, method of lines and pseudo‐spectral methods.

The Fourier pseudo‐spectral method is shown to be the most efficient approach.

This work is original; a computational comparison of numerical schemes applied to liquid crystals has not been found in the literature.

This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third-order nonlinear boundary value problems on the semi-infinite domain.

The approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations.

The method is computationally very attractive and gives very accurate results.

Many problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations.

The purpose of this paper is to propose an approximate method for solving a fractional population growth model in a closed system. The fractional derivatives are described in the Caputo sense.

The approach is based on the differential transform method. The solutions of a fractional model equation are calculated in the form of convergent series with easily computable components.

The diagonal Padé approximants are effectively used in the analysis to capture the essential behavior of the solution.

Illustrative examples are included to demonstrate the validity and applicability of the technique.

The purpose of this paper is to find approximate solutions for a general class of fractional order boundary value problems that arise in engineering applications.

A newly developed semi-analytical scheme will be applied to find approximate solutions for fractional order boundary value problems. The technique is regarded as an extension of the well-established variation iteration method, which was originally proposed for initial value problems, to cover a class of boundary value problems.

It has been demonstrated that the method yields approximations that are extremely accurate and have uniform distributions of error throughout their domain. The numerical examples confirm the method’s validity and relatively fast convergence.

The generalized variational iteration method that is presented in this study is a novel strategy that can handle fractional boundary value problem more effectively than the classical variational iteration method, which was designed for initial value problems.

The purpose of this study is to implement a newly introduced numerical scheme for the numerical solution of a class of nonlinear fractional Bratu-type boundary value problems (BVPs).

This strategy is based on a generalization of the variational iteration method (VIM). This proposed generalized VIM (GVIM) is particularly suitable for tackling BVPs.

This scheme yields accurate solutions for a class of nonlinear fractional Bratu-type BVPs, for which the errors are uniformly distributed across a given domain. A proof of convergence is included. The numerical results confirm that this approach overcomes the deficiency of the VIM and other methods that exist in the literature in the sense that the solution does not deteriorate as the authors move away from the initial starting point.

The method introduced is based on original research that produces new knowledge. To the best of the authors’ knowledge, this is the first time that this GVIM is applied to fractional BVPs.

The purpose of the method is to develop a numerical method for the solution of nonlinear partial differential equations.

A new numerical approach based on Barycentric Rational interpolation has been used to solve partial differential equations.

A numerical technique based on barycentric rational interpolation has been developed to investigate numerical simulation of the Burgers’ and Fisher’s equations. Barycentric interpolation is basically a variant of well-known Lagrange polynomial interpolation which is very fast and stable. Using semi-discretization for unknown variable and its derivatives in spatial direction by barycentric rational interpolation, we get a system of ordinary differential equations. This system of ordinary differential equation’s has been solved by applying SSP-RK43 method. To check the efficiency of the method, computed numerical results have been compared with those obtained by existing methods. Barycentric method is able to capture solution behavior at small values of kinematic viscosity for Burgers’ equation.

To the best of the authors’ knowledge, the method is developed for the first time and validity is checked by stability and error analysis.

A modified version of an essentially non‐oscillatory high resolution scheme is applied to the simulation of two‐dimensional combustion waves. Through the splitting of the governing equations the method is applied to the homogeneous system and the source terms are approximated by means of a Runge‐Kutta third‐order scheme. The results obtained in the modelling of a high speed wave show an improvement with respect to a semi‐implicit scheme in the reconstruction of the shock propagation.