In this paper, we investigate a wide range of dynamical regimes produced by the nonlinearly excited phase (NEP) equation (a single sixthorder nonlinear partial differential equation) using a more advanced numerical method, namely, the integrated radial basis function network method. Previously, we obtained single-step spinning solutions of the equation using the Galerkin method. First, we verify the numerical solver through an exact solution of a forced version of the equation. Doing so, we compare the numerical results obtained for different space and time steps with the exact solution. Then, we apply the method to solve the NEP equation and reproduce the previously obtained spinning regimes. In the new series of numerical experiments, we find regimes in the form of spinning trains of steps/kinks comprising one, two, or three kinks. The evolution of the distance between the kinks is analyzed. Two different kinds of boundary conditions are considered: homogeneous and periodic. The dependence of the dynamics on the size of the domain is explored showing how larger domains accommodate multiple spinning fronts. We determine the critical domain size (bifurcation size) above which non-trivial settled regimes become possible. The initial condition determines the direction of motion of the kinks but not their sizes and velocities.
Fronts of reaction in certain systems (such as so-called solid flames) are modelled by a high-order nonlinear partial differential equation, which we analyse numerically. Previously, Strunin [IMA J. Appl. Math. 63:163-177, 1999] obtained stable spinning solutions of the equation using the Galerkin method. Here we use a more sophisticated and arguably more powerful method, namely the one-dimensional radial basis function method, to study the equation further. As an initial step, we elaborate the numerical code and tested it by reproducing the spinning regimes for a range of initial conditions. In a new series of experiments, we find a regime where the front is shaped as a pair of kinks spinning in a stable joint formation. The settled character of this regime is demonstrated.
In this research work, we implement the 1D-IRBFN method to find approximate solution of a very important non-linear partial differential equation (NLPDE) known as Kuramoto–Sivashinsky equation (KSE). The Crank–Nicolson formulation is used for time integration and the radial basis functions are used for the space discretization for KSE. We use the IRBFN method to solve two versions of KS equation, and then we compare numerical results with the exact solutions to demonstrate the accuracy of the method. The obtained solutions are in good agreement with the exact solutions. To visualize the dynamical behavior of KSE with different parameter values, we show graphical solutions and also compare the numerical solutions with few of results given in the recent research papers. To express the efficiency of the method, we calculate two types of errors in the numerical experiments (a) global relative error and (b) L2 error.Mathematics Subject Classification (2020) 74Sxx · 35G20
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