A novel 1D-FDTD Scheme to Solve the Nonlinear Second-order Thermoviscous Hydrodynamic Model

In this paper, we present a novel and simple Yee Finite-Difference Time-Domain (FDTD) scheme to solve numerically the nonlinear second-order thermoviscous Navier–Stokes and the Continuity equations. In their original form, these equations cannot be discretized by using the Yee’s mesh, at least, easily. As it is known, the use of the Yee’s mesh is recommended because it is optimized in order to obtain higher computational performance and remains at the core of many current acoustic FDTD softwares. In order to use the Yee’s mesh, we propose to rewrite the aforementioned equations in a novel form. To achieve this, we will use the substitution corollary. This procedure is novel in the literature. Although the scheme can be extended to more than one dimension, in this paper, we will focus only on the one-dimensional solution because it can be validated with two analytical solutions to the Burgers equation: the Mendousse mono-frequency solution and the Lardner bi-frequency solution. Numerical solutions are excellently consistent with the analytical solution, which demonstrates the effectiveness of our formulation.This work was partially supported by the “Research Programme for Groups of Scientific Excellence at Region of Murcia” of the Seneca Foundation (Agency for Science and Technology of the Region of Murcia, Spain - 19895/GERM/15). María Campo-Valera is grateful for postdoctoral program Margarita Salas - Spanish Ministry of Universities (financed by European Union - NextGenerationEU)

Non-uniform Haar Wavelet Method for Solving Singularly Perturbed Differential Difference Equations of Neuronal Variability

A non-uniform Haar wavelet method is proposed on specially designed non-uniform grid for the numerical treatment of singularly perturbed differential-difference equations arising in neuronal variability.We convert the delay and shift terms using Taylor series up to second order and then the problem with delay and shift is converted into a new problem without the delay and shift terms. Then it is solved by using non-uniform Haar wavelet. Two test examples have been demonstrated to show the accuracy of the non-uniform Haar wavelet method. The performance of the present method yield more accurate results on increasing the resolution level and converges fast in comparison to uniform Haar wavelet

Non-dyadic Haar Wavelet Algorithm for the Approximated Solution of Higher order Integro-Differential Equations

The objective of this study is to explore non-dyadic Haar wavelets for higher order integro-differential equations. In this research article, non-dyadic collocation method is introduced by using Haar wavelet for approximating the solution of higher order integrodifferential equations of Volterra and Fredholm type. The highest order derivatives in the integrodifferential equations are approximated by the finite series of non-dyadic Haar wavelet and then lower order derivatives are calculated by the process of integration. The integro-differential equations are reduced to a set of linear algebraic equations using the collocation approach. The Gauss - Jordan method is then used to solve the resulting system of equations. To demonstrate the efficiency and accuracy of the proposed method, numerous illustrative examples are given. Also, the approximated solution produced by the proposed wavelet technique have been compared with those of other approaches. The exact solution is also compared to the approximated solution and presented through tables and graphs. For various numbers of collocation points, different errors are calculated. The outcomes demonstrate the effectiveness of the Haar approach in resolving these equations