scholarly journals Well-posedness and numerical approximation of tempered fractional terminal value problems

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Maria Luísa Morgado ◽  
Magda Rebelo
Keyword(s):  
Numerical Methods ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Numerical Approximation ◽  
Shooting Method ◽  
Well Posedness ◽  
Numerical Examples ◽  
Uniqueness Of The Solution ◽  
The Given ◽  
Theoretical Results

AbstractFor a class of tempered fractional terminal value problems of the Caputo type, we study the existence and uniqueness of the solution, analyze the continuous dependence on the given data, and using a shooting method we present and discuss three numerical schemes for the numerical approximation of such problems. Some numerical examples are considered in order to illustrate the theoretical results and evidence the efficiency of the numerical methods.

scholarly journals Identification of the initial temperature from the given temperature data at the left end of a rod

2019 ◽  
Vol 4 (2) ◽  
pp. 469-474
Author(s):  
Yesim Sarac ◽  
S. Sule Sener
Keyword(s):  
Existence And Uniqueness ◽  
Initial Temperature ◽  
Adjoint Problem ◽  
Temperature Data ◽  
Minimizing Sequence ◽  
Fréchet Differential ◽  
Uniqueness Of The Solution ◽  
The Cost ◽  
The Given ◽  
Theoretical Results

AbstractThis study aims to investigate the problem of determining the unknown initial temperature in a variable coefficient heat equation. We obtain the existence and uniqueness of the solution of the optimal control problem considered under some conditions. Using the adjoint problem approach, we get the Frechet differential of the cost functional. We construct a minimizing sequence and give the convergence rate of this sequence. Also, we test the theoretical results in a numerical example by using the MAPLE® program.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals Admissible Hybrid Z-Contractions in b-Metric Spaces

Axioms ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 2 ◽  
Author(s):  
Ioan Cristian Chifu ◽  
Erdal Karapınar
Keyword(s):  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Fixed Point Problem ◽  
Uniqueness Result ◽  
Point Problem ◽  
Well Posedness ◽  
Simulation Functions ◽  
Set Up ◽  
The Given

In this manuscript, we introduce a new notion, admissible hybrid Z -contraction that unifies several nonlinear and linear contractions in the set-up of a b-metric space. In our main theorem, we discuss the existence and uniqueness result of such mappings in the context of complete b-metric space. The given result not only unifies the several existing results in the literature, but also extends and improves them. We express some consequences of our main theorem by using variant examples of simulation functions. As applications, the well-posedness and the Ulam–Hyers stability of the fixed point problem are also studied.

Functions of Minimal Norm with the Given Set of Fourier Coefficients

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 651
Author(s):  
Pyotr Ivanshin
Keyword(s):  
Fourier Series ◽  
Existence And Uniqueness ◽  
Fourier Coefficients ◽  
Nonnegative Function ◽  
Trigonometric Fourier Series ◽  
Minimum Norm ◽  
Minimal Norm ◽  
Uniqueness Of The Solution ◽  
The Given

We prove the existence and uniqueness of the solution of the problem of the minimum norm function ∥ · ∥ ∞ with a given set of initial coefficients of the trigonometric Fourier series c j , j = 0 , 1 , … , 2 n . Then, we prove the existence and uniqueness of the solution of the nonnegative function problem with a given set of coefficients of the trigonometric Fourier series c j , j = 1 , … , 2 n for the norm ∥ · ∥ 1 .

scholarly journals Block-by-Block Method for Solving Nonlinear Volterra-Fredholm Integral Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-8
Author(s):  
Abdallah A. Badr
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Existence And Uniqueness ◽  
Time Dependent ◽  
Volterra Kernel ◽  
Block Method ◽  
Numerical Examples ◽  
Uniqueness Of The Solution ◽  
Fredholm Kernel

We consider a nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind. The Volterra kernel is time dependent, and the Fredholm kernel is position dependent. Existence and uniqueness of the solution to this equation, under certain conditions, are discussed. The block-by-block method is introduced to solve such equations numerically. Some numerical examples are given to illustrate our results.

scholarly journals STABILITY BY HOMOGENIZATION OF THERMOVISCOPLASTIC PROBLEMS

2010 ◽  
Vol 20 (09) ◽  
pp. 1591-1616 ◽  
Author(s):  
NICOLAS CHARALAMBAKIS ◽  
FRANCOIS MURAT
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Strain Rate ◽  
Existence And Uniqueness ◽  
Body Force ◽  
Initial Conditions ◽  
Rate Sensitivity ◽  
Effective Coefficients ◽  
Uniqueness Of The Solution ◽  
Theoretical Results

In this paper, we study the homogenization of the system of partial differential equations describing the quasistatic shearing of heterogeneous thermoviscoplastic materials. We first present the existence and uniqueness of the solution of the above system. We then define "stable by homogenization" models as the models where the equations in both the heterogeneous problems and the homogenized one are of the same form. Finally we show that the model with non-oscillating strain-rate sensitivity which is submitted to steady boundary shearing and body force, is stable by homogenization. In this model, the homogenized (effective) coefficients depend on the initial conditions and on the boundary shearing and body force. Those theoretical results are illustrated by one numerical example.

scholarly journals A LARGE DEFORMATION, VISCOELASTIC, THIN ROD MODEL: DERIVATION AND ANALYSIS

2009 ◽  
Vol 19 (10) ◽  
pp. 1907-1928 ◽  
Author(s):  
J. BEYROUTHY ◽  
H. LE DRET
Keyword(s):  
Existence And Uniqueness ◽  
Finite Strain ◽  
Three Dimensional ◽  
Evolution Problem ◽  
Well Posedness ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Model Derivation ◽  
Regularity Results ◽  
Strain Model

We present a Cosserat-based three-dimensional to one-dimensional reduction in the case of a thin rod, of the viscoelastic finite strain model introduced by Neff. This model is a coupled minimization/evolution problem. We prove the existence and uniqueness of the solution of the reduced minimization problem. We also show a few regularity results for this solution which allow us to establish the well-posedness of the evolution problem. Finally, the reduced model preserves observer invariance.

OPTIMAL SUPERHEDGING UNDER NON-CONVEX CONSTRAINTS — A BSDE APPROACH

2008 ◽  
Vol 11 (04) ◽  
pp. 363-380 ◽  
Author(s):  
CHRISTIAN BENDER ◽  
MICHAEL KOHLMANN
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Numerical Methods ◽  
Borrowing Constraints ◽  
Time Dependent ◽  
Convex Constraints ◽  
Numerical Examples ◽  
Black Scholes ◽  
Backward Sdes ◽  
Theoretical Results

We apply theoretical results by Peng on supersolutions for Backward SDEs (BSDEs) to the problem of finding optimal superhedging strategies in a generalized Black–Scholes market under constraints. Constraints may be imposed simultaneously on wealth process and portfolio. They may be non-convex, time-dependent, and random. The BSDE method turns out to be an extremely useful tool for modeling realistic markets: in this paper, it is shown how more realistic constraints on the portfolio may be formulated via BSDE theory in terms of the amount of money invested, the portfolio proportion, or the number of shares held. Based on recent advances on numerical methods for BSDEs (in particular, the forward scheme by Bender and Denk [1]), a Monte Carlo method for approximating the superhedging price is given, which demonstrates the practical applicability of the BSDE method. Some numerical examples concerning European and American options under non-convex borrowing constraints are presented.

scholarly journals The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Huanting Li ◽  
Yunfei Peng ◽  
Kuilin Wu
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Initial State ◽  
Switching Line ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching <inline-formula><tex-math id="M1">\begin{document}$ k\in \mathbb{N} $\end{document}</tex-math></inline-formula> times on bounded time intervals.</p>

scholarly journals Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

2013 ◽  
Vol 23 (2) ◽  
pp. 341-355 ◽  
Author(s):  
Babak Shiri ◽  
Sedaghat Shahmorad ◽  
Gholamreza Hojjati
Keyword(s):  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Convergence Properties ◽  
Piecewise Continuous ◽  
The Given ◽  
Uniqueness Of A Solution ◽  
Theoretical Results

In this paper, we deal with a system of integral algebraic equations of the Hessenberg type. Using a new index definition, the existence and uniqueness of a solution to this system are studied. The well-known piecewise continuous collocation methods are used to solve this system numerically, and the convergence properties of the perturbed piecewise continuous collocation methods are investigated to obtain the order of convergence for the given numerical methods. Finally, some numerical experiments are provided to support the theoretical results.

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