liu.seSearch for publications in DiVA
Change search
Link to record
Permanent link

Direct link
BETA
Publications (10 of 13) Show all publications
Johansson, B. T. (2017). Konstruktion av solur via vektorer. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Konstruktion av solur via vektorer
2017 (Swedish)Other (Other academic)
Abstract [sv]

En metod baserad på projektion presenteras för att konstruera solur. Formlerna som framtas är generella och täcker olika typerav horisontella solur. De erhållna resultaten är klassiska, men med hjälp av vektorer och projektion fås en sammanhängande framställning. Arbetet är speciellt tänkt som övning i användning av vektorer för de som börjat kurser innehållande vektorräkning.

Place, publisher, year, pages
Linköping: Linköping University Electronic Press, 2017. p. 17
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-140111 (URN)
Available from: 2017-08-30 Created: 2017-08-30 Last updated: 2017-09-07Bibliographically approved
Johansson, T., Lesnic, D. & Reeve, T. (2014). A Mesh less Regularization Method for a Two-Dimensional Two-Phase Linear Inverse Stefan Problem. Advances in Applied Mathematics and Mechanics, 5(6), 825-845
Open this publication in new window or tab >>A Mesh less Regularization Method for a Two-Dimensional Two-Phase Linear Inverse Stefan Problem
2014 (English)In: Advances in Applied Mathematics and Mechanics, ISSN 2070-0733, E-ISSN 2075-1354, Vol. 5, no 6, p. 825-845Article in journal (Refereed) Published
Abstract [en]

In this paper, a meshless regularization method of fundamental solutions is proposed for a two-dimensional, two-phase linear inverse Stefan problem. The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions. Furthermore, the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution. Therefore, regularization is necessary in order to obtain a stable solution. Numerical results for several benchmark test examples are presented and discussed.

Place, publisher, year, edition, pages
GLOBAL SCIENCE PRESS, 2014
Keywords
Heat conduction; method of fundamental solutions (MFS); inverse Stefan problem; two-phase change
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-106819 (URN)10.4208/aamm.2013.m77 (DOI)000334927900004 ()
Available from: 2014-05-28 Created: 2014-05-23 Last updated: 2017-12-05
Johansson, T., Lesnic, D. & Reeve, T. (2014). A meshless method for an inverse two-phase one-dimensional nonlinear Stefan problem. Mathematics and Computers in Simulation, 101, 61-77
Open this publication in new window or tab >>A meshless method for an inverse two-phase one-dimensional nonlinear Stefan problem
2014 (English)In: Mathematics and Computers in Simulation, ISSN 0378-4754, E-ISSN 1872-7166, Vol. 101, p. 61-77Article in journal (Refereed) Published
Abstract [en]

We extend a meshless method of fundamental solutions recently proposed by the authors for the one-dimensional two-phase inverse linear Stefan problem, to the nonlinear case. In this latter situation the free surface is also considered unknown which is more realistic from the practical point of view. Building on the earlier work, the solution is approximated in each phase by a linear combination of fundamental solutions to the heat equation. The implementation and analysis are more complicated in the present situation since one needs to deal with a nonlinear minimization problem to identify the free surface. Furthermore, the inverse problem is ill-posed since small errors in the input measured data can cause large deviations in the desired solution. Therefore, regularization needs to be incorporated in the objective function which is minimized in order to obtain a stable solution. Numerical results are presented and discussed.

Place, publisher, year, edition, pages
Elsevier, 2014
Keywords
Heat conduction; Inverse Stefan problem; Method of fundamental solutions; Two-phase change; Regularization
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-108156 (URN)10.1016/j.matcom.2014.03.004 (DOI)000336693800005 ()
Available from: 2014-06-26 Created: 2014-06-26 Last updated: 2017-12-05
Chapko, R., Johansson, B. & Vavrychuk, V. (2014). Numerical solution of parabolic Cauchy problems in planar corner domains. Mathematics and Computers in Simulation, 101, 1-12
Open this publication in new window or tab >>Numerical solution of parabolic Cauchy problems in planar corner domains
2014 (English)In: Mathematics and Computers in Simulation, ISSN 0378-4754, E-ISSN 1872-7166, Vol. 101, p. 1-12Article in journal (Refereed) Published
Abstract [en]

An iterative method for the parabolic Cauchy problem in planar domains having a finite number of corners is implemented based on boundary integral equations. At each iteration, mixed well-posed problems are solved for the same parabolic operator. The presence of corner points renders singularities of the solutions to these mixed problems, and this is handled with the use of weight functions together with, in the numerical implementation, mesh grading near the corners. The mixed problems are reformulated in terms of boundary integrals obtained via discretization of the time-derivative to obtain an elliptic system of partial differential equations. To numerically solve these integral equations a Nystrom method with super-algebraic convergence order is employed. Numerical results are presented showing the feasibility of the proposed approach.

Place, publisher, year, edition, pages
Elsevier, 2014
Keywords
Heat equation; Cauchy problem; Landweber method; Corner singularities; Boundary integral equation
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-108155 (URN)10.1016/j.matcom.2014.03.001 (DOI)000336693800001 ()
Available from: 2014-06-26 Created: 2014-06-26 Last updated: 2017-12-05
Babenko, C., Chapko, R. & Johansson, T. (2014). On the Numerical Solution of the Laplace Equation with Complete and Incomplete Cauchy Data Using Integral Equations. CMES - Computer Modeling in Engineering & Sciences, 101(5), 299-317
Open this publication in new window or tab >>On the Numerical Solution of the Laplace Equation with Complete and Incomplete Cauchy Data Using Integral Equations
2014 (English)In: CMES - Computer Modeling in Engineering & Sciences, ISSN 1526-1492, E-ISSN 1526-1506, Vol. 101, no 5, p. 299-317Article in journal (Refereed) Published
Abstract [en]

We consider the numerical solution of the Laplace equations in planar bounded domains with corners for two types of boundary conditions. The first one is the mixed boundary value problem (Dirichlet-Neumann), which is reduced, via a single-layer potential ansatz, to a system of well-posed boundary integral equations. The second one is the Cauchy problem having Dirichlet and Neumann data given on a part of the boundary of the solution domain. This problem is similarly transformed into a system of ill-posed boundary integral equations. For both systems, to numerically solve them, a mesh grading transformation is employed together with trigonometric quadrature methods. In the case of the Cauchy problem the Tikhonov regularization is used for the discretized system. Numerical examples are included both for the well-posed and ill-posed cases showing that accurate numerical solutions can be obtained with small computational effort.

Place, publisher, year, edition, pages
TECH SCIENCE PRESS, 2014
Keywords
Laplace equation; Cauchy problem; Corner domain; Mixed problem; Mesh grading transform; Single-layer potential; Tikhonov regularization
National Category
Civil Engineering
Identifiers
urn:nbn:se:liu:diva-114261 (URN)10.3970/cmes.2014.101.299 (DOI)000348259000001 ()
Available from: 2015-02-16 Created: 2015-02-16 Last updated: 2017-12-04
Chapko, R., Johansson, T. & Savka, Y. (2014). On the use of an integral equation approach for the numerical solution of a Cauchy problem for Laplace equation in a doubly connected planar domain. Inverse Problems in Science and Engineering, 22(1), 130-149
Open this publication in new window or tab >>On the use of an integral equation approach for the numerical solution of a Cauchy problem for Laplace equation in a doubly connected planar domain
2014 (English)In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 22, no 1, p. 130-149Article in journal (Refereed) Published
Abstract [en]

We consider the Cauchy problem for the Laplace equation, i.e. the reconstruction of a harmonic function from knowledge of the value of the function and its normal derivative given on a part of the boundary of the solution domain. The solution domain considered is a bounded smooth doubly connected planar domain bounded by two simple disjoint closed curves. Since the Cauchy problem is ill-posed, as a regularizing method we generalize the novel direct integral equation approach in [1], originally proposed for a circular outer boundary curve, to a more general simply connected domain. The solution is represented as a sum of two single-layer potentials defined on each of the two boundary curves and in which both densities are unknown. To identify these densities, the representation is matched with the given Cauchy data to generate a system to solve for the densities. It is shown that the operator corresponding to this system is injective and has dense range, thus Tikhonov regularization is applied to solve it in a stable way. For the discretisation, the Nystrom method is employed generating a linear system to solve, and via Tikhonov regularization a stable discrete approximation to these densities are obtained. Using these one can then find an approximation to the solution of the Cauchy problem. A numerical example is included and we compare with other regularizing methods as well (implemented via integral methods). These results show that the proposed direct method gives accurate reconstructions with little computational effort (the computational time is of order of seconds). Moreover, the obtained approximation can be used as an initial guess in more involved regularizing methods to further improve the accuracy.

Place, publisher, year, edition, pages
Taylor and Francis: STM, Behavioural Science and Public Health Titles, 2014
Keywords
Laplace equation; Nystrom method; single-layer operator; Tikhonov regularization; 35A35
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-103274 (URN)10.1080/17415977.2013.829467 (DOI)000328245900010 ()
Available from: 2014-01-17 Created: 2014-01-16 Last updated: 2017-12-06
Johansson, T., Lesnic, D. & Reeve, T. (2014). The method of fundamental solutions for the two-dimensional inverse Stefan problem. Inverse Problems in Science and Engineering, 22(1), 112-129
Open this publication in new window or tab >>The method of fundamental solutions for the two-dimensional inverse Stefan problem
2014 (English)In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 22, no 1, p. 112-129Article in journal (Refereed) Published
Abstract [en]

We propose an application of the method of fundamental solutions (MFS) for the two-dimensional inverse Stefan problem, where data are to be reconstructed from knowledge of the moving surface and the given Stefan conditions on this surface. We present numerical results for several examples both when the initial data are given but also when it is not specified. These results show good accuracy with low computational cost and are compared with results obtained by other methods.

Place, publisher, year, edition, pages
Taylor and Francis: STM, Behavioural Science and Public Health Titles, 2014
Keywords
heat conduction; method of fundamental solutions (MFS); inverse Stefan problem; Tikhonovs regularization; 35K05; 35A35; 65N35
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-103273 (URN)10.1080/17415977.2013.827180 (DOI)000328245900009 ()
Available from: 2014-01-17 Created: 2014-01-16 Last updated: 2017-12-06
Chapko, R., Johansson, T. & Vavrychuk, V. (2013). A projected iterative method based on integral equations for inverse heat conduction in domains with a cut. Inverse Problems, 29(6), 065003
Open this publication in new window or tab >>A projected iterative method based on integral equations for inverse heat conduction in domains with a cut
2013 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 29, no 6, p. 065003-Article in journal (Refereed) Published
Abstract [en]

The Cauchy problem for the parabolic heat equation, consisting of the reconstruction of the solution from knowledge of the temperature and heat flux on a part of the boundary of the solution domain, is investigated in a planar region containing a cut. This linear inverse ill-posed problem is numerically solved using an iterative regularization procedure, where at each iteration step mixed Dirichlet-Neumann problems for the parabolic heat equation are used. Using the method of Rothe these mixed problems are reduced to a sequence of boundary integral equations. The integral equations have a square root singularity in the densities and logarithmic and hypersingularities in the kernels. Moreover, the mixed parabolic problems have singularities near the endpoints of the cut. Special techniques are employed to handle each of these (four) types of singularities, and analysis is performed in weighted spaces of square integrable functions. Numerical examples are included showing that the proposed regularizing procedure gives stable and accurate approximations.

Place, publisher, year, edition, pages
Institute of Physics: Hybrid Open Access, 2013
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-96123 (URN)10.1088/0266-5611/29/6/065003 (DOI)000320039200003 ()
Available from: 2013-08-14 Created: 2013-08-14 Last updated: 2017-12-06
Reeve, T. & Johansson, T. (2013). The method of fundamental solutions for a time-dependent two-dimensional Cauchy heat conduction problem. Engineering analysis with boundary elements, 37(3), 569-578
Open this publication in new window or tab >>The method of fundamental solutions for a time-dependent two-dimensional Cauchy heat conduction problem
2013 (English)In: Engineering analysis with boundary elements, ISSN 0955-7997, E-ISSN 1873-197X, Vol. 37, no 3, p. 569-578Article in journal (Refereed) Published
Abstract [en]

We investigate an application of the method of fundamental solutions (MFS) to the time-dependent two-dimensional Cauchy heat conduction problem, which is an inverse ill-posed problem. Data in the form of the solution and its normal derivative is given on a part of the boundary and no data is prescribed on the remaining part of the boundary of the solution domain. To generate a numerical approximation we generalize the work for the stationary case in Mann (2011) [23] to the time-dependent setting building on the MFS proposed in Johansson and Lesnic (2008) [15], for the one-dimensional heat conduction problem. We incorporate Tikhonov regularization to obtain stable results. The proposed approach is flexible and can be adjusted rather easily to various solution domains and data. An additional advantage is that the initial data does not need to be known a priori, but can be reconstructed as well.

Place, publisher, year, edition, pages
Elsevier, 2013
Keywords
Heat conduction, Cauchy problem, Inverse problem, Method of fundamental solutions, Regularization
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-92617 (URN)10.1016/j.enganabound.2012.12.008 (DOI)000317314400008 ()
Note

Funding Agencies|EPSRC||

Available from: 2013-05-16 Created: 2013-05-14 Last updated: 2017-12-06
Johansson, T. (2006). An iterative method for a Cauchy problem for the heat equation. IMA Journal of Applied Mathematics, 71(2), 262-286
Open this publication in new window or tab >>An iterative method for a Cauchy problem for the heat equation
2006 (English)In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 71, no 2, p. 262-286Article in journal (Refereed) Published
Abstract [en]

An iterative method for reconstruction of the solution to a parabolic initial boundary value problem of second order from Cauchy data is presented. The data are given on a part of the boundary. At each iteration step, a series of well-posed mixed boundary value problems are solved for the parabolic operator and its adjoint. The convergence proof of this method in a weighted L2-space is included. © 2006. Oxford University Press.

Keywords
Cauchy problem, Heat equation, Iterative regularization method, Mixed problem, Weighted Sobolev space
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-50241 (URN)10.1093/imamat/hxh093 (DOI)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-12
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-9066-7922

Search in DiVA

Show all publications