Shelly Arora’s research while affiliated with Punjabi University and other places

The traveling wave behavior of the nonlinear third and fourth-order advection-diffusion equation has been elaborated. In this study, the effect of dispersion and dissipation processes was mainly analyzed thoroughly. In the thorough analysis, strictly permanent short waves to breaking waves, having comparative higher amplitudes, have been observed. The governed problem was employed with the space-splitting method for a coupled system of equations to conduct the computational process. For the time derivative, the Crank-Nicolson difference approximation was studied. An orthogonal collocation method using Hermite splines has been implemented to approximate the solution of the semi-discretized coupled problem. The proposed method reduces the equation to an iterative scheme of an algebraic system of collocation equations, which reduced the computational complexity. The proposed scheme is found to be unconditionally stable, and the numerical demonstrations and comparisons represented the computational efficiency.

A fully discrete computational technique involving the implicit finite difference technique and cubic Hermite splines is proposed to solve the non-linear conformable damped Burgers’ equation with variable coefficients numerically. The proposed scheme is capable of solving the equation having singularity at t=0. The space direction is discretized using cubic Hermite splines, whereas the time direction is discretized using an implicit finite difference scheme. The convergence, stability and error estimates of the proposed scheme are discussed in detail to prove the efficiency of the technique. The convergence of the proposed scheme is found to be of order h2 in space and order (Δt)α in the time direction. The efficiency of the proposed scheme is verified by calculating error norms in the Eucledian and supremum sense. The proposed technique is applied on conformable damped Burgers’ equation with different initial and boundary conditions and the results are presented as tables and graphs. Comparison with results already in the literature also validates the application of the proposed technique.

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique.

Fourth order extended Fisher Kolmogorov reaction diffusion equation has been solved numerically using a hybrid technique. The temporal direction has been discretized using Crank Nicolson technique. The space direction has been split into second order equation using twice continuously differentiable function. The space splitting results into a system of equations with linear heat equation and non linear reaction diffusion equation. Quintic Hermite interpolating polynomials have been implemented to discretize the space direction which gives a system of collocation equations to be solved numerically. The hybrid technique ensures the fourth order convergence in space and second order in time direction. Unconditional stability has been obtained by plotting the eigen values of the matrix of iterations. Travelling wave behaviour of dependent variable has been obtained and the computed numerical values are shown by surfaces and curves for analyzing the behaviour of the numerical solution in both space and time directions. Mathematics Subject Classification 35B25 · 35C07 · 65D07 · 65M06 · 65N35

Anomalous diffusion of particles in fluids is better described by the fractional diffusion models. A robust hybrid numerical algorithm for a two-dimensional time fractional diffusion equation with the source term is presented. The well-known L1 scheme is considered for semi-discretization of the diffusion equation. To interpolate the semi-discretized equation, orthogonal collocation with bi-quintic Hermite splines as the basis is chosen for the smooth solution. Quintic Hermite splines interpolate the solution as well as its first and second order derivatives. The technique reduces the proposed problem to an algebraic system of equations. Stability analysis of the implicit scheme is studied using $\mathcal {\tilde{H}}_{1}^m$-norm defined in Sobolev space. The optimal order of convergence is found to be of order $O(h^4)$ in spatial direction and is of order $O(\Delta t)^{2-\alpha }$ in the temporal direction where h is the step size in space direction and $\Delta t$ is the step size in time direction and $\alpha$ is the fractional order of the derivative. Numerical illustrations have been presented to discuss the applicability of the proposed hybrid numerical technique to the problems having fractional order derivative.

The present study proposes a hybrid numerical technique to discuss the solution of non-linear reaction-diffusion equations with variable coefficients. The perturbation parameter was assumed to be time-dependent. The spatial domain was discretized using the cubic Hermite splines collocation method. These splines are smooth enough to interpolate the function as well as its tangent at the node points. The temporal domain was discretized using the Crank-Nicolson scheme, commonly known as the CN scheme. The cubic Hermite splines are convergent of order $h^4$, and the CN scheme is convergent of order $\Delta t^2$. The technique is found to be convergent of order $O(h^{2}\big(\gamma_2 \varepsilon_j\Delta t + \gamma_0(1+\bar{\alpha})h^2\big)+\Delta t^2)$. The step size in the space direction is taken to be h , and the step size in the time direction is $\Delta t$. Stability of the proposed scheme was studied using the $L_2$ and $L_{\infty}$ norms. The proposed scheme has been applied to different sets of problems and is found to be more efficient than existing schemes.

Literature review plays a very important role to the success of any research work. It involves perusing works done by other scholars in the research area. It forms the basis for understanding the research topic and helps to bring to light recent works done in the research area and areas still seeking for academic attention. Literature review surveys academic articles, books, and other sources relevant to a precise area of research. These sources provide adequate information to assist a researcher have a good overview of the research area and form an opinion as to which direction his/her research should take. These sources of literature have their different characteristics as well as their individual strengths and weaknesses. In view of this, it is recommended that, the various sources are used to complement each other in a research process. In this article, the authors examine different works done in the area of Radial Basis Functions Networks (RBFs) by reviewing different sources of literature. Radial basis function networks (RBFNs) $\varphi \left( r \right)$ are univariate continuous real valued functions whose output depends on only the distance between a point and a fixed point called centre. They are regarded as a class of feed forward networks with universal approximation capabilities. This makes RBFNs an exciting area in mathematics and thus beginning to receive lots of attention.

An improved collocation technique has been proposed to discretize the fourth-order multi-parameter non-linear Kuramoto -Sivashinsky (K-S) equation. The spatial direction has been discretized with quintic Hermite splines, whereas the temporal direction has been discretized with a weighted finite difference scheme. The fourth-order equation in space direction has been decomposed into second-order using space splitting by introducing a new variable. Space splitting has been proposed to improve the convergence of the approximate solution. The proposed equation has been analyzed on a uniform grid in both space and time directions. Error bounds for general order Hermite splines are established for fully discrete scheme. Stability analysis for the proposed scheme has also been discussed elaborately. Periodic and non periodic problems of K-S equation type have been discussed to study the technique.