Mehdi Dehghan and Vahid Mohammadi
This study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a…
Abstract
Purpose
This study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology.
Design/methodology/approach
First, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations.
Findings
This study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space.
Originality/value
This study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties.
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This bibliography is offered as a practical guide to published papers, conference proceedings papers and theses/dissertations on the finite element (FE) and boundary element (BE…
Abstract
This bibliography is offered as a practical guide to published papers, conference proceedings papers and theses/dissertations on the finite element (FE) and boundary element (BE) applications in different fields of biomechanics between 1976 and 1991. The aim of this paper is to help the users of FE and BE techniques to get better value from a large collection of papers on the subjects. Categories in biomechanics included in this survey are: orthopaedic mechanics, dental mechanics, cardiovascular mechanics, soft tissue mechanics, biological flow, impact injury, and other fields of applications. More than 900 references are listed.
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Sapna Pandit, Pooja Verma, Manoj Kumar and Poonam
This article offered two meshfree algorithms, namely the local radial basis functions-finite difference (LRBF-FD) approximation and local radial basis functions-differential…
Abstract
Purpose
This article offered two meshfree algorithms, namely the local radial basis functions-finite difference (LRBF-FD) approximation and local radial basis functions-differential quadrature method (LRBF-DQM) to simulate the multidimensional hyperbolic wave models and work is an extension of Jiwari (2015).
Design/methodology/approach
In the evolvement of the first algorithm, the time derivative is discretized by the forward FD scheme and the Crank-Nicolson scheme is used for the rest of the terms. After that, the LRBF-FD approximation is used for spatial discretization and quasi-linearization process for linearization of the problem. Finally, the obtained linear system is solved by the LU decomposition method. In the development of the second algorithm, semi-discretization in space is done via LRBF-DQM and then an explicit RK4 is used for fully discretization in time.
Findings
For simulation purposes, some 1D and 2D wave models are pondered to instigate the chastity and competence of the developed algorithms.
Originality/value
The developed algorithms are novel for the multidimensional hyperbolic wave models. Also, the stability analysis of the second algorithm is a new work for these types of model.
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Z. Zaidi, E.H. Twizell, Y. Cherruault and A. Meulemans
To show how the combined Adomian/Alienor methods for solving adaptive control problems can successfully be applied to chemotherapy.
Abstract
Purpose
To show how the combined Adomian/Alienor methods for solving adaptive control problems can successfully be applied to chemotherapy.
Design/methodology/approach
Problem formulation is first developed and combined mathematical methods (Adomian/Alienor) are used for the solution of non‐linear differential equations/systems with unknown parameters and without discretization or linearization. The approach is applied to biological systems and in particular the drug/tumour two compartment model is addressed.
Findings
A general abstract framework for the identification and the control of a non‐linear evolution system has been developed. It was found that it is possible to identify and control a system using a powerful technique based on a combination of the Adomian/Alienor methods. This produced a methodology which showed its superiority over the traditional methods in that as a result of their implementation we can predict and optimize the individual dosage in the described application.
Research limitations/implications
The combined techniques proved to be successful for the optimization of drug administration where account is taken of effectiveness, usefulness and safety. Further research collaboration between multidisciplinary scientists and practitioners directed towards more insight into drug/cancerous cells behaviour is required.
Practical implications
An alternative to other classical techniques to solve control/identification problems has been produced.
Originality/value
New combined technique given which is superior to traditional ones for certain therapeutic cases.
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In this article, the author proposed a new analytical solution procedure for 12 order differential equation. The new analytical technique namely homotopy analysis transform method…
Abstract
Purpose
In this article, the author proposed a new analytical solution procedure for 12 order differential equation. The new analytical technique namely homotopy analysis transform method is efficient and effective for solving higher order differential equations. The results indicate the effectiveness and stability of the proposed algorithm. The paper aims to discuss these issues.
Design/methodology/approach
The author designed a new solution methodology for higher order differential equations which is a combination of Laplace transformation and homotopy deformation theory.
Findings
The author mainly discussed two numerical applications with error analysis of analytical scheme which shows high order accuracy of the projected technique which will serve as a milestone for solving higher order differential equations in engineering with no efforts.
Originality/value
The author proposed an innovative and original idea for nth-order differential equations in this article.
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Ali Saleh Alshomrani, Sapna Pandit, Abdullah K. Alzahrani, Metib Said Alghamdi and Ram Jiwari
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
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).
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Azizeh Jabbari, Hossein Kheiri and Ahmet Yildirim
– The purpose of this paper is to obtain analytic solutions of telegraph equation by the homotopy Padé method.
Abstract
Purpose
The purpose of this paper is to obtain analytic solutions of telegraph equation by the homotopy Padé method.
Design/methodology/approach
The authors used Maple Package to calculate the solutions obtained from the homotopy Padé method.
Findings
The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m, m] homotopy Padé technique are often independent of auxiliary parameter h and this technique accelerates the convergence of the related series. Finally, numerical results for some test problems with known solutions are presented and the numerical results are given to show the efficiency of the proposed techniques.
Originality/value
The paper is shown that homotopy Padé technique is a promising tool with accelerated convergence for complicated nonlinear differential equations.
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The diffusion‐advection phenomena occur in many physical situations such as, the transport of heat in fluids, flow through porous media, the spread of contaminants in fluids and…
Abstract
Purpose
The diffusion‐advection phenomena occur in many physical situations such as, the transport of heat in fluids, flow through porous media, the spread of contaminants in fluids and as well as in many other branches of science and engineering. So it is essential to approximate the solution of these kinds of partial differential equations numerically in order to investigate the prediction of the mathematical models, as the exact solutions are usually unavailable.
Design/methodology/approach
The difficulties arising in numerical solutions of the transport equation are well known. Hence, the study of transport equation continues to be an active field of research. A number of mathematicians have developed the method of time‐splitting to divide complicated time‐dependent partial differential equations into sets of simpler equations which could then be solved separately by numerical means over fractions of a time‐step. For example, they split large multi‐dimensional equations into a number of simpler one‐dimensional equations each solved separately over a fraction of the time‐step in the so‐called locally one‐dimensional (LOD) method. In the same way, the time‐splitting process can be used to subdivide an equation incorporating several physical processes into a number of simpler equations involving individual physical processes. Thus, instead of applying the one‐dimensional advection‐diffusion equation over one time‐step, it may be split into the pure advection equation and the pure diffusion equation each to be applied over half a time‐step. Known accurate computational procedures of solving the simpler diffusion and advection equations may then be used to solve the advection‐diffusion problem.
Findings
In this paper, several different computational LOD procedures were developed and discussed for solving the two‐dimensional transport equation. These schemes are based on the time‐splitting finite difference approximations.
Practical implications
The new approach is simple and effective. The results of a numerical experiment are given, and the accuracy are discussed and compared.
Originality/value
A comparison of calculations with the results of the conventional finite difference techniques demonstrates the good accuracy of the proposed approach.
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Arshad Khan, Mo Faheem and Akmal Raza
The numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear…
Abstract
Purpose
The numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs.
Design/methodology/approach
Because of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods.
Findings
This paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased.
Originality/value
This paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.
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The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods…
Abstract
Purpose
The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods for the solution of the inviscid generalized regularized long-wave (GRLW) equation subject to initial Gaussian conditions.
Design/methodology/approach
Two implicit second- and fourth-order accurate finite difference methods and three Runge-Kutta procedures are introduced. The methods employ a new dependent variable which contains the wave amplitude and its second-order spatial derivative. Numerical experiments are reported for several temporal and spatial step sizes in order to assess their accuracy and the preservation of the first two invariants of the inviscid GRLW equation as functions of the spatial and temporal orders of accuracy, and thus determine the conditions under which grid-independent results are obtained.
Findings
It has been found that the steepening of the wave increase as the nonlinearity exponent is increased and that the accuracy of the fourth-order Runge-Kutta method is comparable to that of a second-order implicit procedure for time steps smaller than 100th, and that only the fourth-order compact method is almost grid-independent if the time step is on the order of 1,000th and more than 5,000 grid points are used, because of the initial steepening of the initial profile, wave breakup and solitary wave propagation.
Originality/value
This is the first study where an accuracy assessment of wave breakup of the inviscid GRLW equation subject to initial Gaussian conditions is reported.