This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. ...It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings.The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.
We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this ...problem in the form of the hybrid conditions. To prove the existence of solutions for our hybrid fractional thermostat equation and inclusion versions, we apply the well-known Dhage fixed point theorems for single-valued and set-valued maps. Finally, we give two examples to illustrate our main results.
The objective of this paper is to investigate, by applying the standard Caputo fractional
-derivative of order
, the existence of solutions for the singular fractional
-integro-differential equation
..., under some boundary conditions where
is singular at some point
, on a time scale
, for
where
and
. We consider the compact map and avail the Lebesgue dominated theorem for finding solutions of the addressed problem. Besides, we prove the main results in context of completely continuous functions. Our attention is concentrated on fractional multi-step methods of both implicit and explicit type, for which sufficient existence conditions are investigated. Finally, we present some examples involving graphs, tables and algorithms to illustrate the validity of our theoretical findings.
By using the fractional Caputo–Fabrizio derivative, we introduce two types new high order derivations called CFD and DCF. Also, we study the existence of solutions for two such type high order ...fractional integro-differential equations. We illustrate our results by providing two examples.
The main classes of inverse problems for equations of mathematical physics and their numerical solution methods are considered in this book which is intended for graduate students and experts in ...applied mathematics, computational mathematics, and mathematical modelling.
In this paper, we study the nonlocal fractional differential equation:
{
D
0
+
α
u
(
t
)
+
f
(
t
,
u
(
t
)
)
=
0
,
0
<
t
<
1
,
u
(
0
)
=
0
,
u
(
1
)
=
η
u
(
ξ
)
,
where
1
<
α
<
2
,
0
<
ξ
<
1
,
η
ξ
α
...−
1
=
1
,
D
0
+
α
is the standard Riemann-Liouville derivative,
f
:
0
,
1
×
0
,
+
∞
)
→
R
is continuous. The existence and uniqueness of positive solutions are obtained by means of the fixed point index theory and iterative technique.
This paper introduces the paradigm of "in-context operator learning" and the corresponding model "In-Context Operator Networks" to simultaneously learn operators from the prompted data and apply it ...to new questions during the inference stage, without any weight update. Existing methods are limited to using a neural network to approximate a specific equation solution or a specific operator, requiring retraining when switching to a new problem with different equations. By training a single neural network as an operator learner, rather than a solution/operator approximator, we can not only get rid of retraining (even fine-tuning) the neural network for new problems but also leverage the commonalities shared across operators so that only a few examples in the prompt are needed when learning a new operator. Our numerical results show the capability of a single neural network as a few-shot operator learner for a diversified type of differential equation problems, including forward and inverse problems of ordinary differential equations, partial differential equations, and mean-field control problems, and also show that it can generalize its learning capability to operators beyond the training distribution.
By mixing the idea of 2-arrays, continued fractions, and Caputo-Fabrizio fractional derivative, we introduce a new operator entitled the infinite coefficient-symmetric Caputo-Fabrizio fractional ...derivative. We investigate the approximate solutions for two infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential problems. Finally, we analyze two examples to confirm our main results.
In this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By ...utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.
We investigate some new class of hybrid type fractional differential equations and inclusions via some nonlocal three-point boundary value conditions. Also, we provide some examples to illustrate our ...results.
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