scholarly journals The Order Completion Method for Systems of Nonlinear PDEs: Solutions of Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Jan Harm van der Walt
Keyword(s):  
Initial Value Problems ◽  
Lower Semicontinuous ◽  
Nonlinear Pdes ◽  
Generalized Solutions ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Initial Condition ◽  
Initial Value ◽  
Extended Sense ◽  
Order Completion

We present an existence result for generalized solutions of initial value problems obtained through the order completion method. The solutions we obtain satisfy the initial condition in a suitable extended sense, and each such solution may be represented in a canonical way through its generalized partial derivatives as nearly finite normal lower semicontinuous function.

scholarly journals Modified Decomposition Method with New Inverse Differential Operators for Solving Singular Nonlinear IVPs in First- and Second-Order PDEs Arising in Fluid Mechanics

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Nemat Dalir
Keyword(s):  
Fluid Mechanics ◽  
Analytical Solutions ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Differential Operators ◽  
Second Order ◽  
Nonlinear Pdes ◽  
Initial Value ◽  
Exact Analytical Solutions ◽  
First Order

Singular nonlinear initial-value problems (IVPs) in first-order and second-order partial differential equations (PDEs) arising in fluid mechanics are semianalytically solved. To achieve this, the modified decomposition method (MDM) is used in conjunction with some new inverse differential operators. In other words, new inverse differential operators are developed for the MDM and used with the MDM to solve first- and second-order singular nonlinear PDEs. The results of the solutions by the MDM together with new inverse operators are compared with the existing exact analytical solutions. The comparisons show excellent agreement.

scholarly journals Finding symmetries by incorporating initial conditions as side conditions

2008 ◽  
Vol 19 (6) ◽  
pp. 701-715 ◽  
Author(s):  
JOANNA GOARD
Keyword(s):  
Initial Value Problems ◽  
Initial Conditions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Initial Condition ◽  
Initial Value ◽  
Side Condition ◽  
Symmetry Generator ◽  
Infinitesimal Symmetry

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

A Characterization of Proximal Subgradient Set-Valued Mappings

1993 ◽  
Vol 36 (1) ◽  
pp. 116-122 ◽  
Author(s):  
R. A. Poliquin
Keyword(s):  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Set Valued Mapping ◽  
Selection Property ◽  
Set Valued Mappings ◽  
Bounded Below

AbstractIn this paper we tackle the problem of identifying set-valued mappings that are subgradient set-valued mappings. We show that a set-valued mapping is the proximal subgradient mapping of a lower semicontinuous function bounded below by a quadratic if and only if it satisfies a monotone selection property.

scholarly journals Smooth variational principles in Radon-Nikodým spaces

1999 ◽  
Vol 60 (1) ◽  
pp. 109-118 ◽  
Author(s):  
Robert Deville ◽  
Abdelhakim Maaden
Keyword(s):  
Banach Space ◽  
Real Function ◽  
Variational Principles ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Weakly Continuous ◽  
Fréchet Differentiable

We prove that if f is a real valued lower semicontinuous function on a Banach space X, for which there exist a > 0 and b ∈ ℝ such that f(x) ≥ 2a∥x∥ + b, x ∈ X, and if X has the Radon-Nikody´m property, then for every Ε > 0 there exists a real function φ X such that φ is Fréchet differentiable, ∥φ∥∞ < Ε, ∥φ′∥∞ < Ε, φ′ is weakly continuous and f + φ attains a minimum on X. In addition, if we assume that the norm in X is β-smooth, we can take the function φ = g1 + g2 where g1 is radial and β-smooth, g2 is Fréchet differentiable, ∥g1∥∞ < Ε, ∥g2∥∞ < Ε, ∥g′1∥∞ < Ε, ∥g′1∥∞ < Ε, g′2 is weakly continuous and f + g1 + g2 attains a minimum on X.

scholarly journals PAIRS OF HAHN AND SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 210-229
Author(s):  
O. Maslyuchenko ◽  
A. Kushnir
Keyword(s):  
Continuous Function ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Normal Space ◽  
Semicontinuous Function ◽  
Upper Semicontinuous ◽  
Pseudocompact Space ◽  
Upper Semicontinuous Function ◽  
Perfectly Normal Space ◽  
Perfectly Normal

In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the infimum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactification any pair of Hahn on X is generated by a separately continuous function on X x Y .

scholarly journals Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 21
Author(s):  
Yasunori Kimura ◽  
Keisuke Shindo
Keyword(s):  
Asymptotic Behavior ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Convergent Sequence ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Geodesic Space ◽  
Proper Convex ◽  
Convex Lower Semicontinuous Function ◽  
Sequence Of Functions

The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To obtain the result, we assume the Mosco convergence of the sets of minimizers of these functions.

scholarly journals Solutions of Smooth Nonlinear Partial Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-37
Author(s):  
Jan Harm van der Walt
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Spatial Dimension ◽  
Nonlinear Pdes ◽  
Generalized Solutions ◽  
Structure Of Solutions ◽  
Partial Differential ◽  
Convergence Spaces ◽  
Order Completion

The method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significant improvement upon the regularity of the solutions and provided new insight into the structure of solutions. In this paper, we show how this method may be adapted so as to allow for the infinite differentiability of generalized functions. Moreover, it is shown that a large class of smooth nonlinear PDEs admit generalized solutions in the space constructed here. As an indication of how the general theory can be applied to particular nonlinear equations, we construct generalized solutions of the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension.

scholarly journals Regularisations of convex functions and slicewise suprema

1994 ◽  
Vol 50 (3) ◽  
pp. 481-499
Author(s):  
S. Simons
Keyword(s):  
Banach Space ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Lower Envelope ◽  
Semicontinuous Function ◽  
Original Function ◽  
Scalar Multiple ◽  
Proper Convex ◽  
Sublinear Functional ◽  
Convex Lower Semicontinuous Function

For a number of years, there has been interest in the regularisation of a given proper convex lower semicontinuous function on a Banach space, defined to be the episum (=inf-convolution) of the function with a scalar multiple of the norm. There is an obvious geometric way of characterising this regularisation as the lower envelope of cones lying above the graph of the original function. In this paper, we consider the more interesting problem of characterising the regularisation in terms of approximations from below, expressing the regularisation as the upper envelope of certain subtangents to the graph of the original function. We shall show that such an approximation is sometimes (but not always) valid. Further, we shall give an extension of the whole procedure in which the scalar multiple of the norm is replaced by a more general sublinear functional. As a by-product of our analysis, we are led to the consideration of two senses stronger than the pointwise sense in which a function on a Banach space can be expressed as the upper envelope of a family of functions. These new senses of suprema lead to some questions in Banach space theorey.

scholarly journals Explicit Solutions of Initial Value Problems for Linear Scalar Riemann-Liouville Fractional Differential Equations With a Constant Delay

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 32 ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Initial Condition ◽  
Constant Delay ◽  
Initial Value ◽  
Fractional Differential ◽  
Set Up ◽  
Ordinary Derivative ◽  
Linear Scalar

In this paper, we study Linear Riemann-Liouville fractional differential equations with a constant delay. The initial condition is set up similarly to the case of ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.

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