Numerical analysis of an ill-posed Cauchy problem for a convection--diffusion equation

Ranjbar, Zohreh; Eldén, Lars

doi:10.1080/17415970600557299

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…From the Figure (14) it can be understood that up to t = 0:2; the noisy data does not a¤ect the heat ‡ux. But after t = 0:2; there is a deviation occurs for all values of and this deviation does not grow further and exhibits unchanging behavior up to t = 1: Particularly for = 0:05, the deviation in the graphs is much more pronounced than the lower values of .…”

Section: Numerical Resultsmentioning

confidence: 88%

“…After that, a number of studies for the inverse problem have been published in [11]- [16]. An inverse forced-convection problem in a channel has been investigated in [11] while convection-di¤usion equation has been solved in [12]- [14]. In addition, advection-di¤usion and reaction-di¤usion equations for an inverse problem case have been considered in [15] and [16], respectively.…”

Section: Introductionmentioning

confidence: 99%

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Implementation of DRBEM for the determination of the heat flux in an inverse problem

Alsoy-Akgün¹

2021

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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A numerical investigation of inverse unsteady natural convection ‡ow in a square cavity …lled with Cu water nano ‡uid is performed. In the direct problem, the enclosure is bounded by one isothermally heated vertical wall at temperature Tm and by three adiabatic walls. In the inverse problem, the enclosure is bounded by right hostile wall on which no boundary condition can be prescribed or measured and by left accessible wall on which both the boundary temperature and heat ‡ux data are overspeci…ed. The dual reciprocity boundary element method (DRBEM) with the fundamental solutions of Laplace and modi…ed Helmholtz equations is used for the solutions of direct and inverse problems. Inhomogeneities are approximated with radial basis functions. Computations are performed for several values of Rayleigh number (Ra), solid volume fraction ( ) and percentage of noise ( ), and accurate and stable results are given for three forms of heat ‡ux namely, steady heat ‡ux (q = q(y)), time dependent uniform heat ‡ux (q = q(t)) and non-uniform time dependent heat ‡ux (q = q(y; t)).

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Section: Numerical Resultsmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Implementation of DRBEM for the determination of the heat flux in an inverse problem

Alsoy-Akgün¹

2021

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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“…The NHDC method is structurally closely related to the inverse heat conduction problems. These were extensively studied and many solution algorithms exists (Eldén 1983;Ranjbar & Eldén 2007). As we are only interested in the relation of time functions between two r-levels, a known upper one and an unknown lower one (Smylie 1965;Benton & Whaler 1983), a solution procedure can be based on solving an integral equation of the first kind.…”

Section: A C K N O W L E D G M E N T Smentioning

confidence: 99%

The 1991 geomagnetic jerk as seen at the Earth's surface and the core-mantle boundary

Ballani

Hagedoorn

Wardinski

et al. 2010

Geophysical Journal International

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S U M M A R YWe determine the occurrence times of geomagnetic impulses (jerks) around the year 1991 in the three geomagnetic secular variation components for the Earth's surface by a simple optimization algorithm. The geomagnetic field models we use are the low-degree parts of the models CM4 and C 3 FM. We find that the temporal jerk pattern can be detected in fields (n ≤ 4), from which the spherical harmonic degrees n = 2 or n = 3 (tangential) and n = 4 (radial) are representative.To calculate the secular variation components at the core-mantle boundary (CMB) we apply the non-harmonic downward continuation (NHDC) method. For the mantle conductivity, three estimates, dependent on the radius, are assumed with conductances between 10 7 S and 2 × 10 8 S. The knowledge of the secular variation components at the CMB allows us to track the global distributions of jerk occurrence times in dependence on the mantle conductivity estimate.We find for each component a typically shaped, global topology for the location of the jerk occurrence times at the Earth's surface and the CMB. For the tangential (ϑ, ϕ)-components, these global topologies show always the well-known temporal bimodality on each surface. Another characteristic feature is found for the jerk of the r-component. It displays a double jerk centred around 1991 consisting of a v-shaped and a reversely v-shaped part, which are significantly correlated.Between the CMB and the Earth's surface, we find time delays in the range of 1-2 yr for the tangential and less than 1 yr for the radial jerk components. To understand these time delays, comparisons at fixed locations are carried out to check the influence of the respective conductivity function. For studying the time delay effects, we apply the inversion set-up of the NHDC to calculate simulated temporal oscillations and derive analytical expressions approximating the phase shifts. We find that jerk occurrence time delays and simulated phase shifts of temporal oscillations have a similar behaviour with respect to the influence of the conductivity and for the radial and the tangential components, respectively.In addition, a new concept for determining a jerk amplitude is presented briefly. This so-called dynamical jerk morphology, which forms a portion of the geomagnetic secular acceleration, is defined for each component by a time function on the considered surface. Its temporal motion patterns at the CMB are likely related with jerk originating processes in the fluid outer core.

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On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

Magnanini

Talenti

2009

Milan J. Math.

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Abstract. Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude -in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage.In the present paper we explore such a principle in dimension 2. We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions. Mathematics Subject Classification (2000). Primary 35F25, 35Q60, 78A05; Secondary 65D25.

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Numerical analysis of an ill-posed Cauchy problem for a convection--diffusion equation

Cited by 4 publications

References 14 publications

Implementation of DRBEM for the determination of the heat flux in an inverse problem

Implementation of DRBEM for the determination of the heat flux in an inverse problem

The 1991 geomagnetic jerk as seen at the Earth's surface and the core-mantle boundary

On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

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