Applications of Solvable Lie Algebras to a Class of Third Order Equations

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed.

Differential model of friction based on bristle model and phase-transition theory

For high-precision position and angle control in robots, it is essential to compensate for the hysteretic behaviour caused by friction when the motion direction is reversed. An accurate friction model which is suitable for control system analysis and implementation is highly desirable. A differential model is proposed in the current paper for the modelling of hysteresis effects caused by friction phenomena. The model is constructed by employing a phenomenological phase-transition theory to mimic the friction mechanism. The bristle friction mechanism is adapted. The switching between static and dynamic friction is regarded as a reversible phase-transition phenomenon, which could be characterized by the local minima of a non-convex potential energy function. The Stribeck effect and the hysteretic relation between friction and velocity are modelled by a nonlinear ordinary differential equation. The comparison of the numerical simulation results and existing experimental friction data are presented. It is illustrated that the friction hysteresis loops are well captured, the capability of the proposed model is verified.

Design of evolutionary cubic spline intelligent solver for nonlinear Painlevé-I transcendent

In this research paper, an innovative bio-inspired algorithm based on evolutionary cubic splines method (CSM) has been utilized to estimate the numerical results of nonlinear ordinary differential equation Painlevé-I. The computational mechanism is used to support the proposed technique CSM and optimize the obtained results with global search technique genetic algorithms (GAs) hybridized with sequential quadratic programming (SQP) for quick refinement. Painlevé-I is solved by the proposed technique CSM-GASQP. In this process, variation of splines is implemented for various scenarios. The CSM-GASQP produces an interpolated function that is continuous upto its second derivative. Also, splines proved to be stable than a single polynomial fitted to all points, and reduce wiggles between the tabulated points. This method provides a reliable and excellent procedure for adaptation of unknown coefficients of splines by searching globally exploiting the performance of GA-SQP algorithms. The convergence, exactness and accuracy of the proposed scheme are examined through the statistical analysis for the several independent runs.

Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation

The article proposes a nonlocal explicit finite-difference scheme for the numerical solution of a nonlinear, ordinary differential equation with a derivative of a fractional variable order of the Gerasimov–Caputo type. The questions of approximation, convergence, and stability of this scheme are studied. It is shown that the nonlocal finite-difference scheme is conditionally stable and converges to the first order. Using the fractional Riccati equation as an example, the computational accuracy of the numerical method is analyzed. It is shown that with an increase in the nodes of the computational grid, the order of computational accuracy tends to unity, i.e., to the theoretical value of the order of accuracy.

APPLICATION OF THE FICTITIOUS DOMAIN METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS

The article shows the ways of applying the method of fictitious domains in solving problems for ordinary differential equations. In the introduction, a small review of the literature on this method, as well as methods for the numerical solution of these problems, is made. The problem statement for the method of fictitious domains for ordinary differential equations is considered. Further, the inequality of estimates was shown. The solution of the auxiliary problem approximates the solution of the original problem with a certain accuracy. The inequality of estimates is obtained in the class of generalized solutions. For the purpose of visual application of the fictitious domain method in problems, a boundary value problem for a one-dimensional nonlinear ordinary differential equation is considered. The problem was written in the form of a difference scheme and led to a solution using the sweep method. In the numerical solution of the problem, numerical calculations were carried out for various values of the parameter included in the auxiliary problem, based on the method of fictitious domains. The numbers of iterations, execution time, and graphs of these calculations are presented and analyzed.

Mathematical analysis and optimal control interventions for sex structured syphilis model with three stages of infection and loss of immunity

In this study, we develop a nonlinear ordinary differential equation to study the dynamics of syphilis transmission incorporating controls, namely prevention and treatment of the infected males and females. We obtain syphilis-free equilibrium (SFE) and syphilis-present equilibrium (SPE). We obtain the basic reproduction number, which can be used to control the transmission of the disease, and thus establish the conditions for local and global stability of the syphilis-free equilibrium. The stability results show that the model is locally asymptotically stable if the Routh–Hurwitz criteria are satisfied and globally asymptotically stable. The bifurcation analysis result reveals that the model exhibits backward bifurcation. We adopted Pontryagin’s maximum principle to determine the optimality system for the syphilis model, which was solved numerically to show that syphilis transmission can be optimally best control using a combination of condoms usage and treatment in the primary stage of infection in both infected male and female populations.

New shape of the chirped bright, dark optical solitons and complex solutions for (2+1) Ginzburg-Landau equation

Investigation of the Ginzburg-Landau equation (GLE) was done to secure new chirped bright, dark periodic and singular function solutions. For this, we used the traveling wave hypothesis and the chirp component. From there it was pointed out the constraint relation to the dierent arbitrary parameters of the GLE. Thereafter, we employed the improved sub-ODE method to handle the nonlinear ordinary differential equation (NODE). It was highlighted the virtue of the used analytical method via new chirped solitary waves. In our knowledge, these results are new, and will be helpful to explain physical phenomenons.

Lie Symmetry Analysis, Conservation Laws, Power Series Solutions, and Convergence Analysis of Time Fractional Generalized Drinfeld-Sokolov Systems

In this work, we investigate invariance analysis, conservation laws, and exact power series solutions of time fractional generalized Drinfeld–Sokolov systems (GDSS) using Lie group analysis. Using Lie point symmetries and the Erdelyi–Kober (EK) fractional differential operator, the time fractional GDSS equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. Moreover, we have constructed conservation laws for time fractional GDSS and obtained explicit power series solutions of the reduced nonlinear ODEs that converge. Lastly, some figures are presented for explicit solutions.