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A large number of diesel engine failures have been reported in the immediate past. The large proportion of these engines that were investigated, were recently overhauled engines that failed soon after the overhaul process. In some cases, these engines failed on the dynamometer, while it was tested before delivery to the customer. The most common failure on a large number of these engines, were pistons seizing in the crown region causing seizure of the piston in the cylinder. Tests were done to correlate the lubricity of the fuel that was used and the failure of the engines. Limits were obtained from which it could be determined when the fuel was not of a proper quality and where engine failures took place. It is finally recommended that the specification SABS 342 be amended to include the requirements for the lubricity of diesel fuels.

This paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number.

In this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. 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 proposed method.

The paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs.

There are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable.

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

The paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.

In this paper, we propose an efficient spectral method for solving the two-dimensional Benjamin–Bona–Mahony–Burgers equation. The new basis functions align well with the problem, the discrete system is sparse and can be efficiently inverted, and the numerical solutions exhibit spectral accuracy in space.

To efficiently simulate the two-dimensional Benjamin–Bona–Mahony–Burgers equation, we utilize transformed generalized Jacobi polynomials and construct the basis functions using the tensor product of these newly introduced polynomials. We provide relevant approximation results. Subsequently, we propose a spectral scheme for the underlying problem, and prove the well-posedness of the scheme, along with the boundedness and energy dissipation of the numerical solutions. We analyze the generalized stability and convergence of the numerical solution of the proposed scheme. Some numerical simulations are presented to demonstrate the efficacy of this newly proposed method.

The new basis functions generated by tensor product of the transformed Jacobi polynomial align well with the underlying problem and simplify the theoretical analysis. The spatial discrete system is sparse and can be efficiently inverted. The numerical solutions exhibit spectral accuracy in space.

We introduce transformed generalized Jacobi polynomials to construct basis functions and present relevant approximation results. We propose an efficient spectral scheme for the two-dimensional Benjamin–Bona–Mahony–Burgers equation, accompanied by optimal error analysis. This new approach achieves spectral accuracy. Moreover, the proposed method and the techniques developed in this work can be applied to simulate a wide range of other nonlinear problems.

Fractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.

This paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.

This paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.

This paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

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.

The purpose of this paper is to introduce a local Newton basis functions collocation method for solving the 2D nonlinear coupled Burgers’ equations. It needs less computer storage and flops than the usual global radial basis functions collocation method and also stabilizes the numerical solutions of the convection-dominated equations by using the Newton basis functions.

A meshless method based on spatial trial space spanned by the local Newton basis functions in the “native” Hilbert space of the reproducing kernel is presented. With the selected local sub-clusters of domain nodes, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. The method leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs).

The method is successfully used for solving the 2D nonlinear coupled Burgers’ equations for reasonably high values of Reynolds number (Re). It is a well-known issue in the analysis of the convection-diffusion problems that the solution becomes oscillatory when the problem becomes convection-dominated if the standard methods are followed without special treatments. In the proposed method, the authors do not detect any instability near the front, hence no technique is needed. The numerical results show that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re.

The authors used more stable basis functions than the standard basis of translated kernels for representing of kernel-based approximants for the numerical solution of partial differential equations (PDEs). The local character of the method, having a well-structured implementation including enforcing the Dirichlet and Neuman boundary conditions, and producing accurate and stable results for flow with reasonably high values of Re for the numerical solution of the 2D nonlinear coupled Burgers’ equations without any special technique are the main values of the paper.

The paper aims to compare and clarify the differences and between the two well-known decomposition spectral techniques; the Winer–Chaos expansion (WCE) and the Winer–Hermite expansion (WHE). The details of the two decompositions are outlined. The difficulties arise when using the two techniques are also mentioned along with the convergence orders. The reader can also find a collection of references to understand the two decompositions with their origins. The geometrical Brownian motion is considered as an example for an important process with exact solution for the sake of comparison. The two decompositions are found practical in analysing the SDEs. The WCE is, in general, simpler, while WHE is more efficient as it is the limit of WCE when using infinite number of random variables. The Burgers turbulence is considered as a nonlinear example and WHE is shown to be more efficient in detecting the turbulence. In general, WHE is more efficient especially in case of nonlinear and/or non-Gaussian processes.

The paper outlined the technical and literature review of the WCE and WHE techniques. Linear and nonlinear processes are compared to outline the comparison along with the convergence of both techniques.

The paper shows that both decompositions are practical in solving the stochastic differential equations. The WCE is found simpler and WHE is the limit when using infinite number of random variables in WCE. The WHE is more efficient especially in case of nonlinear problems.

Applicable for SDEs with square integrable processes and coefficients satisfying Lipschitz conditions.

This paper fulfils a comparison required by the researchers in the stochastic analysis area. It also introduces a simple efficient technique to model the flow turbulence in the physical domain.

The purpose of this paper is to determine both analytically and numerically the kink solutions to a new one-dimensional, viscoelastic generalization of Burgers’ equation, which includes a non-linear constitutive law, and the number of kinks as functions of the non-linearity and relaxation parameters.

An analytical procedure and two explicit finite difference methods based on first-order accurate approximations to the first-order derivatives are used to determine the single- and double-kink solutions.

It is shown that only two parameters characterize the solution and that the existence of a shock wave requires that the (semi-positive) relaxation parameter be less than unity and the non-linearity parameter be less than two. It is also shown that negative values of the non-linearity parameter result in kinks with a single inflection point and strain and dissipation rates with a single relative minimum and a single, relative maximum, respectively. For non-linearity parameters between one and two, it is shown that the kink has three inflection points that merge into a single one as this parameter approaches one and that the strain and dissipation rates exhibit relative maxima and minima whose magnitudes decrease and increase as the relaxation and nonlinearity coefficients, respectively, are increased. It is also shown that the viscoelastic generalization of the Burgers equation presented here is related to an ϕ⁸−scalar field.

A new, one-dimensional, viscoelastic generalization of Burgers’ equation, which includes a non-linear constitutive law and relaxation is proposed, and its kink solutions are determined both analytically and numerically. The equation and its solutions are connected with scalar field theories and may be used to both studies the effects of the non-linearity and relaxation and assess the accuracy of numerical methods for first-order, non-linear partial differential equations.

The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation.

The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree.

The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program.

There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.