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In the present paper, two effective numerical schemes depending on a second-order Strang splitting technique are presented to obtain approximate solutions of the one-dimensional Burgers’ equation utilizing the collocation technique and approximating directly the solution by multiquadric-radial basis function (MQ-RBF) method. To show the performance of both schemes, we have considered two examples of Burgers’ equation. The obtained numerical results are compared with the available exact values and also those of other published methods. Moreover, the computed $$L_{2}$$ and $$L_{\infty }$$ error norms have been given. It is found that the presented schemes produce better results as compared to those obtained almost all the schemes present in the literature.

In this paper, a numerical scheme based on quintic B-spline collocation method using the Strang splitting technique is presented for solving the generalised Rosenau–regularised long wave (RLW) equation given by appropriate initial-boundary values. For this purpose, firstly the problem is split into two subproblems such that each one includes the derivative in the direction of time. Secondly, each subproblem is reduced to a system of ordinary differential equations (ODEs) using collocation finite-element method with quintic B-splines for spatial integration. Then, the resulting ODEs for time integration are solved using the Strang time-splitting technique with the second order via the usual Runge–Kutta (RK-4) algorithm with the fourth order. To measure the accuracy and efficiency of the present scheme, a model problem with an exact solution is taken into consideration and investigated for various values of the parameter p. The error norms $$L_{2}$$ and $$L_{\infty }$$ together with the invariants of discrete mass Q and discrete energy E have been computed and a comparison is given with other ones found in the literature. The convergence order of the present numerical scheme has also been computed. Furthermore, the stability analysis of the scheme is numerically examined.

Obtaining exact solutions of the evolution equation is one of the very important subjects in mathematics, science and technology. For this purpose, many different methods have been constructed and developed. In this article, a new technique which is called extended modified Exp(-$\Omega (\xi )$) function method is going to be studied for seeking new exact solutions of Burger-Fisher equation and Phi Four equation. The method is capable of deriving many number of solutions. With the aid of the method, various exact solutions including trigonometric, hyperbolic and rational solutions have been obtained and using a software the graphical representation of the solutions have been presented. In conclusion, we can say that the present method can also be used for the solutions of a wide range of problems.

In this article, the generalized Rosenau–KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie–Trotter and the second-order Strang time-splitting techniques combined with the quintic B-spline collocation by the help of the fourth order Runge–Kutta (RK-4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L2 and L? with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated. © 2019 Wiley Periodicals, Inc.

In this paper, a numerical solution of the modified Burgers equation is obtained by a cubic B-spline collocation method. In the solution process, a linearization technique based on quasi-linearization has been applied to deal with the non-linear term appearing in the equation. The computed results are compared with others selected from the available literature. The error norms $$L_{2}$$ and $$L_{\infty }$$ are computed and found to be sufficiently small. A Fourier stability analysis of the method is also investigated.

In this paper, numerical solutions of a coupled modified Korteweg–de Vries equation have been obtained by the quadratic B-spline Galerkin finite element method. The accuracy of the method has been demonstrated by five test problems. The obtained numerical results are found to be in good agreement with the exact solutions. A Fourier stability analysis of the method is also investigated.

In this paper, a coupled Burgers’ equation has been numerically solved by a Galerkin quadratic B-spline FEM. The performance of the method has been examined on three test problems. Results obtained by the method have been compared with known exact solution and other numerical results in the literature. A Fourier stability analysis of the method is also investigated. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, the G ′ G -expansion method is applied to the Liouville, sine–Gordon and new coupled MKdV equations to obtain their some generalized exact travelling wave solutions.

In this paper we use a linearized numerical scheme based on finite difference method to obtain solitary wave solutions of the one-dimensional modified equal width (MEW) equation. Two test problems including the motion of a single solitary wave and the interaction of two solitary waves are solved to demonstrate the efficiency of the proposed numerical scheme. The obtained results show that the proposed scheme is an accurate and efficient numerical technique in the case of small space and time steps. A stability analysis of the scheme is also investigated.

A key driver of thermistor response is a thermally-dependent electrical conductivity. Several variants have been proposed in the literature and it is the purpose of this paper to explore the applicability of a bulk model by way of using the heat balance integral approach to predict the response of a positive temperature coefficient thermistor. Conclusions are presented regarding the sensitivity of the response to electrical conductivity.

A linearized implicit finite-difference method is presented to find numerical solutions of the equal width wave equation. The method has been used successfully to investigate the motion of a single solitary wave, the development of the interaction of two solitary waves and an undular bore. The obtained results are compared with other numerical results in the literature. A stability analysis of the scheme is also investigated.

In this paper, a lumped Galerkin method based on quadratic B-spline finite elements is used to find numerical solutions of the one-dimensional regularized long wave (RLW) equation with a variant of initial and boundary conditions. The obtained numerical results show that the present method is a remarkably successful numerical technique for solving the equation. Results are compared with published numerical solutions. A linear stability analysis of the scheme is also investigated.

Variable space grid and boundary immobilisation schemes based on the explicit finite difference method are applied to the one-phase Stefan-like problems with a forcing term in order to evaluate the temperature distribution and the interface movement (location and speed). The numerical results obtained by the two schemes have been compared with the exact values and are found to be in good agreement with each other. It is shown that the numerical solutions exhibit the expected convergence to the exact one as the mesh size is reduced. A von-Neumann stability analysis of each scheme is also investigated.

In this paper, a Galerkin quadratic B-spline finite element method is used to solve numerically the one-dimensional Burgers' equation reduced by the Hopf-Cole transformation. The performance of the method is tested on two model problems involving moderately Reynolds numbers with known exact solutions. The obtained numerical results show that the method is efficient, robust and reliable for solving Burgers' equation accurately even involving high Reynolds numbers for which the exact solution fails. Computed results are compared with other numerical results in the literature. A stability analysis of the method is also investigated.

In this paper, an analytical-numerical method is applied to the one-dimensional Korteweg-de Vries equation with a variant of boundary and initial conditions to obtain its numerical solutions at small times. Two test problem with known exact solutions are studied to demonstrate the accuracy of the present method. The obtained results are compared with the exact solution of each problem and are found to be in good agreement with each other. The numerical scheme is also compared with earlier work and shown to be accurate and efficient.

In this paper, subdomain collocation (SC) and Petrov-Galerkin (PG) methods based on spline finite elements have been applied to the one-dimensional positive temperature coefficient (PTC) thermistor problem with a temperature dependent ramp function electrical conductivity to obtain the predicted temperature distributions and the locations of the free boundaries in the steady-state case. The numerical results obtained by the present methods have been compared with the exact solution and also those obtained by earlier authors. A Fourier stability analysis of each method is investigated.

This paper presents approximate steady-state solutions of a one-dimensional positive temperature coefficient (PTC) thermistor problem, having a step function electrical conductivity that is a highly non-linear function of the temperature, using subdomain collocation and Petrov-Galerkin methods based on spline finite elements. The resulting system of ordinary differential equations is solved by the usual Crank-Nicolson finite difference method using a variant of Thomas algorithm. It is shown that the numerical solutions obtained by the present methods exhibit the correct physical characteristics of the problem and, they are in very good agreement with the exact solution. A Fourier stability analysis of each method is also investigated.

This paper presents approximate steady-state solutions of a one-dimensional positive temperature coefficient thermistor problem with a ramp function electrical conductivity using the Galerkin cubic B-spline finite element method. The numerical results obtained by the present method have been compared with the exact one and also those obtained by earlier authors, and are found to be in very good agreement with each other. Further, it is shown that the numerical solutions satisfy the correct physical phenomena of the problem.

A numerical solution of the one-dimensional Burgers equation is obtained using a lumped Galerkin method with quadratic B-spline finite elements. The scheme is implemented to solve a set of test problems with known exact solutions. Results are compared with published numerical and exact solutions. The proposed scheme performs well. A linear stability analysis shows the scheme to be unconditionally stable. E-mail: aesen@inonu.edu.tr

A linearized implicit finite-difference method is presented to find numerical solutions of the one-dimensional Burgers-like equations. The method has been used successfully to obtain accurate numerical solutions even for small values of viscosity term @n. Results obtained by the present method using a direct technique for some values of @n have been compared with the exact values and are found to be in good agreement with each other.

In this paper, approximate steady-state solutions of a one-dimensional positive temperature coefficient thermistor problem with a modified step function electrical conductivity are obtained by using the Galerkin cubic B-spline finite element method. It is shown that the computational results obtained by the method display the correct physical characteristics of the problem, and they are found to be in very good agreement with the exact solution. It is also shown that the numerical solution exhibits the expected convergence to the exact one as the mesh size is refined. Further a Fourier stability analysis of the method is investigated.

In this study, a least-squares quadratic B-spline finite element method for calculating the numerical solutions of the one-dimensional Burgers-like equations with appropriate boundary and initial conditions is presented. Three test problems have been studied to demonstrate the accuracy of the present method. Results obtained by the method have been compared with the exact solution of each problem and are found to be in good agreement with each other. A Fourier stability analysis of the method is also investigated.

In this paper, we present a numerical scheme based on an isotherm migration formulation for one-dimensional, one-phase Stefan problem with a time dependent Neumann condition on the fixed boundary and a constant Dirichlet condition on the moving boundary. The numerical results obtained by the present method have been compared with exact one and also those obtained by earlier authors, and are found to be in very good agreement with each other. It is also shown that the numerical solution displays the expected convergence to the exact one as the mesh size is refined.

This paper presents approximate steady-state solutions of a positive temperature coefficient thermistor problem, having a ramp electrical conductivity that is a highly non-linear function of the temperature, using a standard explicit finite difference method. It is shown that numerical solutions exhibit the correct physical characteristics of the problem and, they are in good agreement with the exact solution.

In this paper, the enthalpy method based on suitable finite difference approximations has been applied to the one-dimensional moving boundary problem with a Neumann-type boundary condition known as the Stefan problem. The numerical results obtained by the hopscotch technique are compared with the exact solution of the problem. It is shown that all results are found to be in very good agreement with each other, and the numerical solution displays the expected convergence to the exact one as the mesh size is refined.

This manuscript is going to seek travelling wave solutions of some coupled partial differential equations with an expansion method known as Sine- Gordon expansion method. Primarily, we are going to employ a wave transformation to partial differential equation to reduce the equations into ordinary differential equations. Then, the solution form of the handled equations is going to be constructed as polynomial of hyperbolic trig or trig functions. Finally, with the aid of symbolic computation, new exact solutions of the partial differentials equations will have been found.

In this paper a heat balance integral (HBI) method is applied to the one-dimensional non-linear Korteweg-de Vries (KdV) equation prescribed by appropriate homogenous boundary conditions and a set of initial conditions to obtain its approximate analytical solutions at small times. It is shown that the HBI solutions obtained by the method may be used effectively at small times when the exact solution of the KdV equation is not known.

We consider the numerical solution of a one-dimensional thermistor (thermo-electric) problem with a new modified step function electrical conductivity which is an inherently non-linear function of the temperature. A variety of finite difference methods are applied to solve the problem using a new modification of the step function electrical conductivity to be satisfied the physical phenomena of the problem.

In this paper, a quadratic B-spline Galerkin finite element approach is applied to one-dimensional coupled KdV equation in order to obtain its numerical solutions. The performance of the method is examined on three test problems. Computed results are compared with the exact results and also other numerical results given in the literature. A Fourier stability analysis of the approach is also done.

A simplified model of one particular thermistor is presented and an approximate functional solution for the evolving coupled thermo-electric problem is obtained by an application of the heat balance integral method. The solution is shown to exhibit all the correct physical characteristics and this simple model extracts a mechanism by which the observed cracking of these devices might be initiated.

A linearized implicit finite difference method to obtain numerical solution of the one-dimensional regularized long-wave (RLW) equation is presented. The performance and the accuracy of the method are illustrated by solving three test examples of the problem: a single solitary wave, two positive solitary waves interaction, and an undular bore. The obtained results are presented and compared with earlier work.

This paper presents finite-difference solution and analytical solution of the finite-difference approximations based on the standard explicit method to the one-dimensional Burgers equation which arises frequently in the mathematical modelling used to solve problems in fluid dynamics. Results obtained by these ways for some modest values of viscosity have been compared with the exact (Fourier) one. It is shown that they are in good agreement with each other.

In this paper, variable space grid and boundary Immobilisation Techniques based on the explicit finite difference are applied to the one-phase classical Stefan problem. It is shown that all the results obtained by the two methods are in good agreement with the exact solution, and exhibit the expected convergence as the mesh size is refined.

This paper applies He's Exp-function method to the one-dimensional general improved KdV (GIKdV) equation with n th power nonlinear term to obtain some new generalized solitary solutions and periodic solutions. It is shown that the Exp-function method, with the help of any symbolic computation packages, provides a straightforwar d and powerful mathematical tool for solving many generalized nonlinear evolution equations arising in mathematical physics.