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In this study, novel numerical methods are presented for solving nonlinear Fisher equations. These equations have a wide range of applications in various scientific and engineering fields, particularly in the biomedical sciences for determining the size of brain tumors. The challenges posed by the nonlinearity of the equations are effectively addressed through the development of numerical techniques. The nonlinearity is tackled using a combination of the method of lines and backward differentiation formulas of varied orders. This method is unconditionally stable, and its accuracy is evaluated using error norms. The methods are successfully validated against test problems with known solutions, demonstrating their superiority through comparative analyses with existing methodologies in the literature.

In this paper, a high-order accurate numerical method for two-dimensional coupled hyperbolic and parabolic partial differential system arising in unsteady magnetohydrodynamics (MHD) flow of upper-convected Maxwell fluid is presented. We apply Galerkin-Legendre spectral method for discretizing spatial derivatives for MHD-M (MHD flow of Maxwell fluid) equations and then Crank-Nicolson scheme is used for time derivatives. Optimal a priori error bound is derived in the L 2 and H 1 norm for the semidiscrete formulation. We then proceed to discuss an instability mechanism in polynomial spectral methods and show that filtering suffices to ensure stability. Optimal error bound is derived between the semi-analytical solutions and filtered semi-analytical solutions. The results are illustrated by computational experiments. Furthermore, these results indicate that the proposed method is simple, accurate, efficient, and stable for solving various MHD-M and MHD flow problems or in general coupled hyperbolic-parabolic partial differential equations. • New MHD equations are derived for upper-convected Maxwell fluid. • These new equations are studied. • Filtering technique introduced and applied. [ABSTRACT FROM AUTHOR]

The motive of this work is to develop "-uniform numerical method for solving singularly perturbed parabolic delay differential equation with small delay. To approximate the term with the delay, Taylor series expansion is used. The resulting singularly perturbed parabolic differential equation is solved by using non-standard finite difference method in spatial direction and implicit Runge-Kutta method for the resulting system of IVPs in temporal direction. Theoretically the developed method is shown to be accurate of order O(N-1 + (t)2) by preserving "-uniform convergence. Two numerical examples are considered to investigate "- uniform convergence of the proposed scheme and the result obtained agreed with the theoretical one. [ABSTRACT FROM AUTHOR]

A uniformly convergent numerical method is developed for solving singularly perturbed 1-D parabolic convection-diffusion problems. The developed method applies a non-standard finite difference method for the spatial derivative discretization and uses the implicit Runge-Kutta method for the semi-discrete scheme. The convergence of the method is analyzed, and it is shown to be first order convergent. To validate the applicability of the proposed method two model examples are considered and solved for different perturbation parameters and mesh sizes. The numerical and experimental results agree well with the theoretical findings. [ABSTRACT FROM AUTHOR]

The radial basis functions (RBFs) depend on an auxiliary parameter, called the shape parameter. Great theoretical and numerical efforts have been made to find the relationship between the accuracy of the RBF-approximations and the value of the shape parameter. In many cases, the numerical approaches are based on minimization of an estimation of the error function, such as Rippa's approach and its modifications. These approaches determine a value for the shape parameter. In this paper, we propose a practical approach to determine an interval, instead of a value, without any minimization and estimation of error function. The idea is based on adding a loop on the shape parameter, but not to minimize an error norm. Suitable values of the shape parameter are determined by take into account the practical convergence behavior of the problem. The proposed method is applied to some illustrative examples, including one and two-dimensional interpolation and partial differential equations. The results of the method are compared with those of some other approaches to confirm the reliability of the method. Furthermore, numerical stability of the method of line respect to the shape parameter is considered for time-dependent PDEs. The results show that, the proposed method can be used as an independent method to find a shape parameter, and also as an alternative for investigating the validity of the values of the other methods. [ABSTRACT FROM AUTHOR]

We propose a simple and accurate numerical scheme for solving the time fractional telegraph (TFT) equation within Caputo type fractional derivative. A fictitious coordinate ϑ is imposed onto the problem in order to transform the dependent variable u ( x , t ) into a new variable with an extra dimension. In the new space with the added fictitious dimension, a combination of method of line and group preserving scheme (GPS) is proposed to find the approximate solutions. This method preserves the geometric structure of the problem. Power and accuracy of this method has been illustrated through some examples of TFT equation. [ABSTRACT FROM AUTHOR]

Fragmentation of polymer-bridged flocs is commonly encountered in many areas of suspension processing such as solid-liquid separation, washing and clarification. The effect of high shear impact on the fragmentation of polyelectrolyte-bridged silica flocs was investigated. The monodisperse silica suspension was first flocculated and then subjected to prolonged breakage in a shear field until a constant size distribution was attained. These flocs were then subjected to high shear impact breakage. The evolution of particle size distribution vs shearing time was monitored. This temporal evolution of size density function was compared with that obtained via population balance modelling converted to logarithmic scale to counter the 6-12 decades in particle size in volume. The Method of Lines (MoL) was employed to solve the equation. At lower breakage speed of 1500 rpm, excellent agreement between experimental and model size density functions was obtained where excellent matching was observed with the first and second moments. However, at the higher speed of 2500 rpm, the agreement is less satisfactory particularly with the variance. Scanning electron microscope images showed physical breakage of particles at this high shear rate. © 2015 Curtin University of Technology and John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A new 2D and 3D software, using the method of line, has been proposed in this paper to analyze planar tunable microwave devices including ferroelectric thin-films. Next, a dynamic characterization method of ferroelectric layers and some first experimental results have been presented. [ABSTRACT FROM AUTHOR]

In this work a powerful method is presented to solve the Klein–Gordon equation (KGE). Firstly, a discretization is implemented on the original equation. Then a geometric technique is applied to obtain the approximate solutions. Some examples containing one-dimensional and two-dimensional are solved to prove the power and the capability of the offered scheme. Indeed, graphs of the exact solution, numerical solution, absolute error and the contour plot of error are provided successfully. [ABSTRACT FROM AUTHOR]