Search results

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.

The purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging.

A conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers.

The transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions.

A new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.

The purpose of this paper is to implement the Ritz‐Galerkin method in Bernstein polynomial basis to give approximation solution of a parabolic partial differential equation with non‐local boundary conditions.

The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then the Ritz‐Galerkin method is utilized to reduce the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

The authors applied the method presented in this paper and solved three test problems.

This is the first time that the Ritz‐Galerkin method in Bernstein polynomial basis is employed to solve the model investigated in the current paper.

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

The main aim of this paper is to use the cubic B‐spline scaling functions for the numerical solution of the 1D parabolic inverse problem with temperature overspecification.

A 1D parabolic partial differential equation with given initial condition and known boundary conditions and subject to overspecification at a point in the spatial domain is solved using the cubic B‐spline scaling functions.

In this paper, a numerical technique is developed for solving an inverse parabolic problem.

The scheme described is efficient. The numerical results obtained using the new algorithm for a test problem show that the new method can solve the model effectively.

The problem investigated in this research has already been studied by several authors. However, to the best of one's knowledge this is the first time that the new approach has been employed to solve the given model.

The purpose of this study is to provide an insight and to solve numerically the identification of an unknown coefficient of radiation/absorption/perfusion appearing in the heat equation from additional temperature measurements.

First, the uniqueness of solution of the inverse coefficient problem is briefly discussed in a particular case. However, the problem is still ill-posed as small errors in the input data cause large errors in the output solution. For numerical discretization, the finite difference method combined with a regularized nonlinear minimization is performed using the MATLAB toolbox routine lsqnonlin.

Numerical results presented for three examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.

The mathematical formulation is restricted to identify coefficients which separate additively in unknown components dependent individually on time and space, and this may be considered as a research limitation. However, there is no research implication to overcome this, as the known input data are also limited to single measurements of temperature at a particular time and space location.

As noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise.

The identification of the additive time- and space-dependent perfusion coefficient will be of great interest to the bio-heat transfer community and applications.

The current investigation advances previous studies which assumed that the coefficient multiplying the lower-order temperature term depends on time or space separately. The knowledge of this physical property coefficient is very important in biomedical engineering for understanding the heat transfer in biological tissues. The originality lies in the insight gained by performing for the first time numerical simulations of inversion to find the coefficient additively dependent on time and space in the heat equation from noisy measurements.

This paper seeks to develop an efficient B‐spline Galerkin scheme for solving the Fisher's equation, which is a nonlinear reaction diffusion equation describing the relation between the diffusion and nonlinear multiplication of a species.

The solution domain is partitioned into uniform mesh and, using the quartic B‐spline functions, the Galerkin method is applied to the Fisher's equation.

The method yields stable accurate solutions. Obtained results are acceptable and in unison with some earlier studies.

Using the uniform mesh, quartic B‐spline Galerkin method is employed for finding the numerical solutions of Fisher's equation.

The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients.

The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well.

Several test problems are provided to confirm the validity and efficiently of the proposed method.

For the first time, some famous examples are solved by using the proposed high-order technique.

The purpose of this paper is to introduce a novel approach based on the high-order matrix derivative of the Bernstein basis and collocation method and its employment to solve an interesting and ill-posed model in the heat conduction problems, homogeneous backward heat conduction problem (BHCP).

By using the properties of the Bernstein polynomials the problems are reduced to an ill-conditioned linear system of equations. To overcome the unstability of the standard methods for solving the system of equations an efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-condition system.

The presented numerical results through table and figures demonstrate the validity and applicability and accuracy of the technique.

A novel method based on the high-order matrix derivative of the Bernstein basis and collocation method is developed and well-used to obtain the numerical solutions of an interesting and ill-posed model in heat conduction problems, homogeneous BHCP with high accuracy.