scholarly journals Riccati PDEs That Imply Curvature-Flatness

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 537
Author(s):  
Iulia Hirica ◽  
Constantin Udriste ◽  
Gabriel Pripoae ◽  
Ionel Tevy
Keyword(s):  
Differential Geometry ◽  
Partial Differential Equations ◽  
Projection Operator ◽  
Point Of View ◽  
Vector Spaces ◽  
Partial Differential ◽  
The Third ◽  
Kronecker Delta ◽  
Symmetric Connection ◽  
Differential Relations

In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in the cases of both symmetric and non-symmetric connection. Although the curvature-flatness idea is classic for symmetric connection, our main theorems about flatness solutions are completely new, leaving for a while the point of view of differential geometry and entering that of PDEs. The second goal is to introduce and study some strong partial differential relations associated to curvature-flatness. The third goal is to introduce and analyze some vector spaces of exotic objects that change the meaning of a generalized Kronecker delta projection operator, in order to discover new PDEs implying curvature-flatness. Significant examples clarify some ideas.

Barycentric Jacobi Spectral Method for Numerical Solutions of the Generalized Burgers-Huxley Equation

2017 ◽  
Vol 18 (1) ◽  
pp. 67-81 ◽  
Author(s):  
Edson Pindza ◽  
M. K. Owolabi ◽  
K.C. Patidar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Point Of View ◽  
Partial Differential ◽  
Computational Point ◽  
Exponential Time Differencing ◽  
Predictor Corrector ◽  
Numerical Approaches

AbstractNumerical solutions of nonlinear partial differential equations, such as the generalized and extended Burgers-Huxley equations which combine effects of advection, diffusion, dispersion and nonlinear transfer are considered in this paper. Such system can be divided into linear and nonlinear parts, which allow the use of two numerical approaches. Barycentric Jacobi spectral (BJS) method is employed for the spatial discretization, the resulting nonlinear system of ordinary differential equation is advanced with a fourth-order exponential time differencing predictor corrector. Comparative numerical results for the values of options are presented. The proposed method is very elegant from the computational point of view. Numerical computations for a wide variety of problems, show that the present method offers better accuracy and efficiency in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.

CHAOTIC BEHAVIOR AND CHAOS CONTROL FOR A CLASS OF COMPLEX PARTIAL DIFFERENTIAL EQUATIONS

2001 ◽  
Vol 12 (06) ◽  
pp. 889-899 ◽  
Author(s):  
G. M. MAHMOUD ◽  
H. A. ABDUSALAM ◽  
A. A. M. FARGHALY
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fixed Points ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Point Of View ◽  
Partial Differential ◽  
Ginzburg Landau ◽  
Practical Point ◽  
Nagumo Equations

Systems of complex partial differential equations, which include the famous nonlinear Schrödinger, complex Ginzburg–Landau and Nagumo equations, as examples, are important from a practical point of view. These equations appear in many important fields of physics. The goal of this paper is to concentrate on this class of complex partial differential equations and study the fixed points and their stability analytically, the chaotic behavior and chaos control of their unstable periodic solutions. The presence of chaotic behavior in this class is verified by the existence of positive maximal Lyapunov exponent.The problem of chaos control is treated by applying the method of Pyragas. Some conditions on the parameters of the systems are obtained analytically under which the fixed points are stable (or unstable).

scholarly journals The Third Boundary Value Problem for a Loaded Thermal Conductivity Equation with a Fractional Caputo Derivative

2020 ◽  
pp. 52-64
Author(s):  
M. Kh. Beshtokov ◽  
M. Z. Khudalov
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Caputo Derivative ◽  
Boundary Value ◽  
Conductivity Equation ◽  
Partial Differential ◽  
The Third ◽  
Fractional Caputo Derivative

Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional (porous media) orders are widely used, since analytical solving methods for solving are impossible.In the paper, we study the initial-boundary value problem for the loaded differential heat equation with a fractional Caputo derivative and conditions of the third kind. To solve the problem on the assumption that there is an exact solution in the class of sufficiently smooth functions by the method of energy inequalities, a priori estimates are obtained both in the differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. Due to the linearity of the problem under consideration, these inequalities allow us to state the convergence of the approximate solution to the exact solution at a rate equal to the approximation order of the difference scheme. An algorithm for the numerical solution of the problem is constructed.

scholarly journals Improvement of the Modified Decomposition Method for Handling Third-Order Singular Nonlinear Partial Differential Equations with Applications in Physics

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Nemat Dalir
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Differential Operators ◽  
Nonlinear Partial Differential Equations ◽  
Initial Value ◽  
Third Order ◽  
Partial Differential ◽  
The Third

The modified decomposition method (MDM) is improved by introducing new inverse differential operators to adapt the MDM for handling third-order singular nonlinear partial differential equations (PDEs) arising in physics and mechanics. A few case-study singular nonlinear initial-value problems (IVPs) of third-order PDEs are presented and solved by the improved modified decomposition method (IMDM). The solutions are compared with the existing exact analytical solutions. The comparisons show that the IMDM is effectively capable of obtaining the exact solutions of the third-order singular nonlinear IVPs.

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