•Time-dependent FC-72 flow boiling heat transfer in minichannels.•11 rectangular 1 mm deep channels with smooth and enhanced heated wall surface.•Detection of temperature on the heated wall by ...infrared thermography.•Two inverse Cauchy-type problems solved by methods using T-functions.•Results: temperature, heat transfer coefficient, uncertainty analysis, boiling curves.
This paper focuses on time-dependent flow boiling heat transfer. The main element of the research stand was the test section with 11 rectangular minichannels of 1 mm depth. The FC-72 was heated by a thin foil during upward flow. The temperature of the outer surface of the foil was measured by infrared thermography. Two types of foil surface were tested: smooth and enhanced with recesses produced using emery paper. The main objective was to verify whether the enhanced surface helps to obtain a better heat transfer efficiency compared to the smooth surface. The 2D mathematical model of heat transfer was proposed, while methods with Trefftz functions were applied to solve the time-dependent problem of heat transfer. The Robin condition was used to determine heat transfer coefficients at the interface between the foil and the fluid. Experimental data on temperature and pressure and their validation were presented and discussed. Results based on the heat transfer coefficients were provided. They covered the calculation procedure, the determination of the mean relative error and results validation with using the 1D approach. The heat transfer coefficients as a consequence of application of the Trefftz method and the hybrid Picard-Trefftz method in the calculation procedure with comparison analysis of the results were shown. Furthermore, comparisons aimed at obtaining the results characteristic for smooth and enhanced surfaces were also conducted. In the experiment with an enhanced heated surface, the values of the heat transfer coefficient appeared to be higher. Boiling curves were illustrated and discussed.
By using Moreau's decomposition theorem for projecting onto cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is ...shown that Picard's method applied to the system of equations associated with the problem of projecting onto a simplicial cone generates a sequence that converges linearly to the solution of the system. Numerical experiments are presented making the comparison between Picard's and semi-smooth Newton's methods to solve the nonsmooth system associated with the problem of projecting a point onto a simplicial cone.
This communication deals with the analytical solutions of Cauchy problem for Cauchy-Riemann system of equations which is basically unstable according to Hadamard but its solution exists if its ...initial data is analytic. Here we used the Vectorial Adomian Decomposition (VAD) method, Vectorial Variational Iteration (VVI) method, and Vectorial Modified Picard’s Method (VMP) method to get the convergent series solution. These suggested schemes give analytical approximation in an infinite series form without using discretization. These methods are effectual and reliable which is demonstrated through six model problems having variety of source terms and analytic initial data.
•Two numerical methods for the distributed order Linear fractional differential equations.•Both methods solve problems without conversion into multi-term Linear fractional DE.•IVPs are solved by ...combination of Caputo definition and Chebyshev operation matrix.•BVPs are solved by a controlled Picard's method with an auxiliary parameter.•Several numerical examples are verified with better accuracy than existing methods.
This paper introduces two methods for the numerical solution of distributed order linear fractional differential equations. The first method focuses on initial value problems (IVPs) and based on the αth Caputo fractional definition with the shifted Chebyshev operational matrix of fractional integration. By applying this method, the IVPs are converted into simple linear differential equations which can be easily handled. The other method focuses on boundary value problems (BVPs) based on Picard's method frame. This method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series. Several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods. Also, the response of mechanical system described by such equations is studied.
We consider the solution of systems of nonlinear algebraic equations that appear in a positivity preserving finite volume scheme for steady-state advection-diffusion equations. We propose and analyze ...numerically an efficient strategy for accelerating the Picard method when it is applied to these systems. The strategy is based on the Anderson acceleration and the adaptive inexact solution of linear systems. We demonstrate its numerical robustness for three black-box preconditioners. PUBLICATION ABSTRACT
Recently, Wilmer III and Costa introduced a method into the mathematics education research literature which they employed to construct solutions to certain classes of ordinary differential equations. ...In this article, we build on their ideas in the following ways. We establish a link between their approach and the method of successive approximations. We show how applying the method of approximations naturally leads to the constructed approximation of Wilmer III and Costa. The new link is important because it enables us to respond to several challenges posed by Wilmer III and Costa. This includes addressing issues raised therein with convergence of their recursively constructed sequence of functions, and responding to their call regarding more mathematical rigour when relaxing the polynomial condition on the coefficients in the differential equation. Furthermore, the new link is pedagogically significant because it also opens up new pedagogical points of view. For example, the results in this paper provide potentially alternate, but overlapping, perspectives that are suitable for, and jointly inform, the learning and teaching of solution methods to differential equations. The value of this is supported by the presumption of Tisdell that teaching multiple ways to solve the same problem has academic and social value.
We discuss the Abel–Gontscharoff boundary value problem on a measure chain in two aspects, namely, the existence and uniqueness of solutions, and the convergence of iterative methods which include ...quasilinearization, approximate quasilinearization, Picard's and approximate Picard's methods. Examples are also presented to dwell upon the importance of the results obtained.
Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem sup
τ
E
(max
0≤
t
≤τ
X
t
−
c
τ), where
X
= (
X
t
)
t
≥0
is geometric Brownian motion with ...drift μ and volatility σ > 0, and the supremum is taken over all stopping times for
X
. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ
*
= inf {
t
> 0 |
X
t
=
g
*
(max
0≤
t
≤
s
X
s
)} where
s
↦
g
*
(
s
) is the
maximal
solution of the (nonlinear) differential equation
under the condition 0 <
g
(
s
) <
s
, where Δ = 1 − 2μ / σ
2
and
K
= Δ σ
2
/ 2
c
. The estimate is established
g
*
(
s
) ∼ ((Δ − 1) /
K
Δ)
1 / Δ
s
1−1/Δ
as
s
→ ∞. Applying these results we prove the following maximal inequality:
where τ may be any stopping time for
X
. This extends the well-known identity
E
(sup
t
>0
X
t
) = 1 − (σ
2
/ 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.
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