GLIMPSES OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN THE TWENTIETH CENTURY: A PRIORI ESTIMATES AND THE BERNSTEIN PROBLEM

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

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A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

Advances in Difference Equations ◽

10.1186/s13662-020-03049-2 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Said Mesloub ◽

Hassan Eltayeb Gadain

Keyword(s):

Differential Equations ◽

Evolution Equation ◽

A Priori Estimates ◽

Boundary Value ◽

A Priori ◽

Initial Boundary ◽

A Priori Bounds ◽

Fractional Evolution Equation ◽

Priori Estimates ◽

Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

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Analytical mathematical schemes: Circular rod grounded via transverse Poisson’s effect and extensive wave propagation on the surface of water

Open Physics ◽

10.1515/phys-2020-0163 ◽

2020 ◽

Vol 18 (1) ◽

pp. 545-554

Author(s):

Asghar Ali ◽

Aly R. Seadawy ◽

Dumitru Baleanu

Keyword(s):

Wave Propagation ◽

Partial Differential Equations ◽

Differential Equations ◽

Longitudinal Wave ◽

Symbolic Computation ◽

Boussinesq Equation ◽

Nonlinear Partial Differential Equations ◽

Wave Model ◽

Simple Equation ◽

Partial Differential

AbstractThis article scrutinizes the efficacy of analytical mathematical schemes, improved simple equation and exp(-\text{Ψ}(\xi ))-expansion techniques for solving the well-known nonlinear partial differential equations. A longitudinal wave model is used for the description of the dispersion in the circular rod grounded via transverse Poisson’s effect; similarly, the Boussinesq equation is used for extensive wave propagation on the surface of water. Many other such types of equations are also solved with these techniques. Hence, our methods appear easier and faster via symbolic computation.

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