{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:28:51Z","timestamp":1772119731651,"version":"3.50.1"},"reference-count":65,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:00:00Z","timestamp":1690156800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:00:00Z","timestamp":1690156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006254","name":"Ruhr-Universit\u00e4t Bochum","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006254","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Multidim Syst Sign Process"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We report on a novel method for solving one-dimensional moving boundary problems based on wave digital principles. Here, we exploit multidimensional wave digital algorithms to derive an efficient and robust algorithm for the solution of the considered problem. Our method lets the wave digital model, on which the algorithm is based, expand according to the size of the solution domain. The expanding model introduces new dynamical elements, which must be properly initialized to obtain a calculable algorithm. To deal with this problem, we make use of linear multistep methods to extrapolate the missing values. Our results show the proposed method to indeed be capable of correctly solving a one-dimensional partial differential equation describing a growing biological axon.<\/jats:p>","DOI":"10.1007\/s11045-023-00881-z","type":"journal-article","created":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T17:01:33Z","timestamp":1690218093000},"page":"703-730","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Solving a one-dimensional moving boundary problem based on wave digital principles"],"prefix":"10.1007","volume":"34","author":[{"given":"Bakr","family":"Al\u00a0Beattie","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karlheinz","family":"Ochs","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,24]]},"reference":[{"key":"881_CR1","doi-asserted-by":"crossref","unstructured":"Alexiades, V. (1993). Mathematical modeling of melting and freezing processes (1st ed.). Routledge.","DOI":"10.1115\/1.2930032"},{"key":"881_CR2","doi-asserted-by":"crossref","unstructured":"Balemarthy, K., & Bass, S. C. (1995). General, linear boundary conditions in MD wave digital simulations. In 1995 IEEE international symposium on circuits and systems (ISCAS), vol.\u00a01, pp. 73\u201376.","DOI":"10.1109\/ISCAS.1995.521454"},{"issue":"2","key":"881_CR3","doi-asserted-by":"publisher","first-page":"694","DOI":"10.1109\/TCSI.2018.2867508","volume":"66","author":"A Bernardini","year":"2019","unstructured":"Bernardini, A., Werner, K. J., Smith, J. O., & Sarti, A. (2019). Generalized wave digital filter realizations of arbitrary reciprocal connection networks. IEEE Transactions on Circuits and Systems I: Regular Papers, 66(2), 694\u2013707. https:\/\/doi.org\/10.1109\/TCSI.2018.2867508","journal-title":"IEEE Transactions on Circuits and Systems I: Regular Papers"},{"issue":"3","key":"881_CR4","doi-asserted-by":"publisher","first-page":"1269","DOI":"10.1109\/TCSI.2020.3044002","volume":"68","author":"A Bernardini","year":"2021","unstructured":"Bernardini, A., Maffezzoni, P., & Sarti, A. (2021). Vector wave digital filters and their application to circuits with two-port elements. IEEE Transactions on Circuits and Systems I: Regular Papers, 68(3), 1269\u20131282. https:\/\/doi.org\/10.1109\/TCSI.2020.3044002","journal-title":"IEEE Transactions on Circuits and Systems I: Regular Papers"},{"key":"881_CR5","doi-asserted-by":"crossref","unstructured":"Bilbao, S. (2004). 3. In Multidimensional wave digital networks. Wiley; pp. 53\u2013113. Available from: https:\/\/onlinelibrary.wiley.com\/doi\/abs\/10.1002\/0470870192.ch3.","DOI":"10.1002\/0470870192.ch3"},{"issue":"7","key":"881_CR6","doi-asserted-by":"publisher","first-page":"1321","DOI":"10.1109\/TCSI.2004.830687","volume":"51","author":"NK Bose","year":"2004","unstructured":"Bose, N. K., & Fettweis, A. (2004). Skew symmetry and orthogonality in the equivalent representation problem of a time-varying multiport inductor. IEEE Transactions on Circuits and Systems I: Regular Papers, 51(7), 1321\u20131329. https:\/\/doi.org\/10.1109\/TCSI.2004.830687","journal-title":"IEEE Transactions on Circuits and Systems I: Regular Papers"},{"issue":"3","key":"881_CR7","doi-asserted-by":"publisher","first-page":"601","DOI":"10.2514\/1.39301","volume":"23","author":"TJ Bright","year":"2009","unstructured":"Bright, T. J., & Zhang, Z. M. (2009). Common misperceptions of the hyperbolic heat equation. Journal of Thermophysics and Heat Transfer, 23(3), 601\u2013607. https:\/\/doi.org\/10.2514\/1.39301","journal-title":"Journal of Thermophysics and Heat Transfer"},{"key":"881_CR8","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1126\/science.1114394","volume":"310","author":"TH Bullock","year":"2005","unstructured":"Bullock, T. H., Bennett, M. V. L., Johnston, D., Josephson, R. K., Marder, E., & Fields, R. D. (2005). The Neuron Doctrine. Redux. Science, 310, 791\u2013793.","journal-title":"Science"},{"issue":"4","key":"881_CR9","first-page":"431","volume":"247","author":"M Cataneo","year":"1958","unstructured":"Cataneo, M. (1958). Sur une forme de l\u2019equation de la chaleur eliminant le paradoxe d\u2019une propagation instantane. Comptes Rendus Academie des Sciences Paris, 247(4), 431\u2013433.","journal-title":"Comptes Rendus Academie des Sciences Paris"},{"key":"881_CR10","doi-asserted-by":"publisher","first-page":"2013","DOI":"10.1103\/PhysRev.131.2013","volume":"131","author":"M Chester","year":"1963","unstructured":"Chester, M. (1963). Second sound in solids. Physical Review, 131, 2013\u20132015. https:\/\/doi.org\/10.1103\/PhysRev.131.2013","journal-title":"Second sound in solids. Physical Review"},{"issue":"1","key":"881_CR11","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/BF01963532","volume":"3","author":"GG Dahlquist","year":"1963","unstructured":"Dahlquist, G. G. (1963). A special stability problem for linear multistep methods. BIT Numerical Mathematics, 3(1), 27\u201343. https:\/\/doi.org\/10.1007\/BF01963532","journal-title":"BIT Numerical Mathematics"},{"issue":"4","key":"881_CR12","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1002\/cta.4490230412","volume":"23","author":"M Erbar","year":"1995","unstructured":"Erbar, M., & Horneber, E. H. (1995). Models for transmission lines with connecting transistors based on wave digital filters. International Journal of Circuit Theory and Applications, 23(4), 395\u2013412. https:\/\/doi.org\/10.1002\/cta.4490230412","journal-title":"International Journal of Circuit Theory and Applications"},{"key":"#cr-split#-881_CR13.1","doi-asserted-by":"crossref","unstructured":"Fettweis, A. & Basu, S. (2015). Modelling of multidimensional (MD) heat diffusion via the Kirchhoff paradigm. In 2015 IEEE international symposium on circuits and systems (ISCAS)","DOI":"10.1109\/ISCAS.2015.7169161"},{"key":"#cr-split#-881_CR13.2","unstructured":"Fettweis, A. & Basu, S. (2015). Modelling of multidimensional"},{"key":"#cr-split#-881_CR13.3","unstructured":"(MD) heat diffusion via the Kirchhoff paradigm. In 2015 IEEE international symposium on circuits and systems (ISCAS); pp. 2373-2376."},{"key":"881_CR14","doi-asserted-by":"crossref","unstructured":"Fettweis, A. (1991). The role of passivity and losslessness in multidimensional digital signal processing-new challenges. In 1991 IEEE international sympoisum on circuits and systems, vol. 1; pp. 112\u2013115.","DOI":"10.1109\/ISCAS.1991.176286"},{"issue":"3","key":"881_CR15","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1002\/jnm.1660050307","volume":"5","author":"A Fettweis","year":"1992","unstructured":"Fettweis, A. (1992). Multidimensional wave digital filters for discrete-time modelling of Maxwell\u2019s equations. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 5(3), 183\u2013201. https:\/\/doi.org\/10.1002\/jnm.1660050307","journal-title":"International Journal of Numerical Modelling: Electronic Networks, Devices and Fields"},{"key":"881_CR16","unstructured":"Fettweis, A. (1994). Multidimensional wave-digital principles: From filtering to numerical integration. In Proceedings of ICASSP \u201994 IEEE international conference on acoustics, speech and signal processing, vol. 6, vi:VI\/173\u2013VI\/181."},{"key":"881_CR17","doi-asserted-by":"crossref","unstructured":"Fettweis, A. (2002). Improved wave-digital approach to numerically integrating the PDES of fluid dynamics. In 2002 IEEE International symposium on circuits and systems. Proceedings (Cat. No.02CH37353). vol.\u00a03; p. III.","DOI":"10.1109\/ISCAS.2002.1010235"},{"issue":"2","key":"881_CR18","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1109\/PROC.1986.13458","volume":"74","author":"A Fettweis","year":"1986","unstructured":"Fettweis, A. (1986). Wave digital filters: Theory and practice. Proceedings of the IEEE, 74(2), 270\u2013327.","journal-title":"Proceedings of the IEEE"},{"issue":"1","key":"881_CR19","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1007\/s11045-005-6236-3","volume":"17","author":"A Fettweis","year":"2006","unstructured":"Fettweis, A. (2006). Robust numerical integration using wave-digital concepts. Multidimensional Systems and Signal Processing, 17(1), 7\u201325. https:\/\/doi.org\/10.1007\/s11045-005-6236-3","journal-title":"Multidimensional Systems and Signal Processing"},{"issue":"1","key":"881_CR20","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s11045-010-0135-y","volume":"22","author":"A Fettweis","year":"2011","unstructured":"Fettweis, A., & Basu, S. (2011). Multidimensional causality and passivity of linear and nonlinear systems arising from physics. Multidimensional Systems and Signal Processing, 22(1), 5\u201325. https:\/\/doi.org\/10.1007\/s11045-010-0135-y","journal-title":"Multidimensional Systems and Signal Processing"},{"issue":"1\u20132","key":"881_CR21","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1007\/BF00927832","volume":"3","author":"A Fettweis","year":"1991","unstructured":"Fettweis, A., & Nitsche, G. (1991). Numerical integration of partial differential equations using principles of multidimensional wave digital filters. Journal of VLSI Signal Processing Systems for Signal, Image and Video Technology, 3(1\u20132), 7\u201324.","journal-title":"Journal of VLSI Signal Processing Systems for Signal, Image and Video Technology"},{"issue":"2","key":"881_CR22","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1007\/BF01938221","volume":"2","author":"A Fettweis","year":"1991","unstructured":"Fettweis, A., & Nitsche, G. (1991). Transformation approach to numerically integrating PDEs by means of WDF principles. Multidimensional Systems and Signal Processing, 2(2), 127\u2013159. https:\/\/doi.org\/10.1007\/BF01938221","journal-title":"Multidimensional Systems and Signal Processing"},{"issue":"5","key":"881_CR23","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1078\/1434-8411-54100120","volume":"56","author":"D Fr\u00e4nken","year":"2002","unstructured":"Fr\u00e4nken, D., & Ochs, K. (2002). Improving wave digital simulation by extrapolation techniques. AEU - International Journal of Electronics and Communications, 56(5), 327\u2013336.","journal-title":"AEU - International Journal of Electronics and Communications"},{"issue":"20","key":"881_CR24","doi-asserted-by":"publisher","first-page":"11589","DOI":"10.1073\/pnas.96.20.11589","volume":"96","author":"JA Galbraith","year":"1999","unstructured":"Galbraith, J. A., Reese, T. S., Schlief, M. L., & Gallant, P. E. (1999). Slow transport of unpolymerized tubulin and polymerized neurofilament in the squid giant axon. Proceedings of the National Academy of Sciences, 96(20), 11589\u201311594.","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"9","key":"881_CR25","doi-asserted-by":"publisher","first-page":"1757","DOI":"10.1016\/j.cma.2006.09.016","volume":"196","author":"H G\u00f3mez","year":"2007","unstructured":"G\u00f3mez, H., Colominas, I., Navarrina, F., & Casteleiro, M. (2007). A finite element formulation for a convection-diffusion equation based on Cattaneo\u2019s law. Computer Methods in Applied Mechanics and Engineering, 196(9), 1757\u20131766. https:\/\/doi.org\/10.1016\/j.cma.2006.09.016","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"issue":"17","key":"881_CR26","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1007\/s11831-010-9042-5","volume":"06","author":"H Gomez","year":"2010","unstructured":"Gomez, H., Colominas, I., Navarrina, F., Par\u00eds, J., & Casteleiro, M. (2010). A hyperbolic theory for advection-diffusion problems: Mathematical foundations and numerical modeling. Archives of Computational Methods in Engineering, 06(17), 191\u2013211.","journal-title":"Archives of Computational Methods in Engineering"},{"issue":"1","key":"881_CR27","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/s10827-006-5330-3","volume":"20","author":"BP Graham","year":"2006","unstructured":"Graham, B. P., Lauchlan, K., & Mclean, D. R. (2006). Dynamics of outgrowth in a continuum model of neurite elongation. Journal of Computational Neuroscience, 20(1), 43.","journal-title":"Journal of Computational Neuroscience"},{"key":"881_CR28","first-page":"1","volume-title":"In Stiff problems\u2013One-step methods","author":"E Hairer","year":"1991","unstructured":"Hairer, E., & Wanner, G. (1991). In Stiff problems\u2013One-step methods (pp. 1\u2013254). Berlin, Heidelberg: Springer."},{"key":"881_CR29","doi-asserted-by":"crossref","unstructured":"Hemetsberger, G., & Hellfajer, R. (1994). Approach to simulating acoustics in supersonic flow by means of multidimensional vector-WDFs. In 1994 IEEE International Symposium on Circuits and Systems (ISCAS), vol.\u00a05; pp. 73\u201376.","DOI":"10.1109\/ISCAS.1994.409303"},{"key":"881_CR30","doi-asserted-by":"crossref","unstructured":"Hetmanczyk, G. & Ochs, K. (2009). Initialization of linear multistep methods in multidimensional wave digital models. In 2009 52nd IEEE international midwest symposium on circuits and systems; pp. 786\u2013789.","DOI":"10.1109\/MWSCAS.2009.5235886"},{"issue":"24","key":"881_CR31","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1002\/jnm.768","volume":"03","author":"G Hetmanczyk","year":"2011","unstructured":"Hetmanczyk, G., & Ochs, K. (2011). A practical guide to multidimensional wave digital models using the example of fluid dynamics. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 03(24), 154\u2013174. https:\/\/doi.org\/10.1002\/jnm.768","journal-title":"International Journal of Numerical Modelling: Electronic Networks, Devices and Fields"},{"key":"881_CR32","doi-asserted-by":"publisher","unstructured":"Higuchi, T. (1963). Mechanism of sustained-action medication. Theoretical analysis of rate of release of solid drugs dispersed in solid matrices. Journal of Pharmaceutical Sciences.,52(12), 1145\u20131149. https:\/\/doi.org\/10.1002\/jps.2600521210.","DOI":"10.1002\/jps.2600521210"},{"key":"881_CR33","doi-asserted-by":"crossref","unstructured":"Jenderny, S., & Ochs, K. (2022). Wave digital emulation of a bio-inspired circuit for axon growth. In 2022 IEEE Biomedical Circuits and Systems Conference (BioCAS); p. 260\u2013264.","DOI":"10.1109\/BioCAS54905.2022.9948613"},{"key":"881_CR34","doi-asserted-by":"crossref","unstructured":"Lawson, S. S., & Guzman, J. G. (2001). On the modelling of the 2D wave equation using multidimensional wave digital filters. In ISCAS 2001. The 2001 IEEE international symposium on circuits and systems (Cat. No.01CH37196), vol.\u00a02; p. 377\u2013380","DOI":"10.1109\/ISCAS.2001.921086"},{"key":"881_CR35","doi-asserted-by":"publisher","unstructured":"Lee, P. I. (2011). Modeling of drug release from matrix systems involving moving boundaries: Approximate analytical solutions. International Journal of Pharmaceutics, 418(1), 18\u201327. https:\/\/doi.org\/10.1016\/j.ijpharm.2011.01.019. Mathematical modeling of drug delivery systems","DOI":"10.1016\/j.ijpharm.2011.01.019"},{"key":"881_CR36","doi-asserted-by":"crossref","unstructured":"Leuer, C. & Ochs, K. (2009). Systematic derivation of reference circuits for wave digital modeling of passive linear partial differential equations. In 2009 52nd IEEE international midwest symposium on circuits and systems; pp. 782\u2013785.","DOI":"10.1109\/MWSCAS.2009.5235885"},{"issue":"7","key":"881_CR37","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1002\/cta.752","volume":"40","author":"C Leuer","year":"2012","unstructured":"Leuer, C., & Ochs, K. (2012). On systematic wave digital modeling of passive hyperbolic partial differential equations. International Journal of Circuit Theory and Applications, 40(7), 709\u2013731. https:\/\/doi.org\/10.1002\/cta.752","journal-title":"International Journal of Circuit Theory and Applications"},{"issue":"54","key":"881_CR38","doi-asserted-by":"publisher","first-page":"321","DOI":"10.2528\/PIER05021002","volume":"01","author":"I Lindell","year":"2005","unstructured":"Lindell, I. (2005). Electromagnetic wave equation in differential-form representation. Progress in Electromagnetics Research, 01(54), 321\u2013333. https:\/\/doi.org\/10.2528\/PIER05021002","journal-title":"Progress in Electromagnetics Research"},{"issue":"19","key":"881_CR39","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1002\/jnm.616","volume":"07","author":"K Luhmann","year":"2006","unstructured":"Luhmann, K., & Ochs, K. (2006). A novel interpretation of the hyperbolization method used to solve the parabolic neutron diffusion equations by means of the wave digital concept. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 07(19), 345\u2013364.","journal-title":"International Journal of Numerical Modelling: Electronic Networks, Devices and Fields"},{"issue":"2048","key":"881_CR40","doi-asserted-by":"publisher","first-page":"2437","DOI":"10.1098\/rspa.2004.1288","volume":"460","author":"DR McLean","year":"2004","unstructured":"McLean, D. R., & Graham, B. P. (2004). Mathematical formulation and analysis of a continuum model for tubulin-driven neurite elongation. Proceedings of the Royal Society of London Series A, 460(2048), 2437\u20132456.","journal-title":"Proceedings of the Royal Society of London Series A"},{"issue":"4","key":"881_CR41","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1109\/MCAS.2018.2872664","volume":"18","author":"K Meerk\u00f6tter","year":"2018","unstructured":"Meerk\u00f6tter, K. (2018). On the passivity of wave digital networks. IEEE Circuits and Systems Magazine, 18(4), 40\u201357. https:\/\/doi.org\/10.1109\/MCAS.2018.2872664","journal-title":"IEEE Circuits and Systems Magazine"},{"key":"881_CR42","doi-asserted-by":"crossref","unstructured":"Michaelis D, Ochs K, Beattie BA, & Jenderny S. (2022). Towards A self-organizing neuronal network based on guided axon-growth. In 2022 IEEE 65th international Midwest symposium on circuits and systems (MWSCAS); pp. 1\u20134.","DOI":"10.1109\/MWSCAS54063.2022.9859299"},{"issue":"3","key":"881_CR43","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1006\/jtbi.1996.0355","volume":"186","author":"KE Miller","year":"1997","unstructured":"Miller, K. E., & Samuels, D. C. (1997). The axon as a metabolic compartment: Protein degradation, transport, and maximum length of an axon. Journal of Theoretical Biology, 186(3), 373\u2013379.","journal-title":"Journal of Theoretical Biology"},{"issue":"98","key":"881_CR44","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/S0169-328X(01)00290-X","volume":"02","author":"E Morrison","year":"2002","unstructured":"Morrison, E., Moncur, P., & Askham, J. (2002). EB1 identifies sites of microtubule extension during neurite formation. Brain Research Molecular Brain Research, 02(98), 145\u201352.","journal-title":"Brain Research Molecular Brain Research"},{"issue":"3","key":"881_CR45","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1078\/1434-8411-00024","volume":"55","author":"K Ochs","year":"2001","unstructured":"Ochs, K. (2001). Passive integration methods: Fundamental theory. AEU - International Journal of Electronics and Communications, 55(3), 153\u2013163. https:\/\/doi.org\/10.1078\/1434-8411-00024","journal-title":"AEU - International Journal of Electronics and Communications"},{"key":"881_CR46","volume-title":"Theorie zeitvarianter linearer \u00dcbertragungssysteme","author":"K Ochs","year":"2012","unstructured":"Ochs, K. (2012). Theorie zeitvarianter linearer \u00dcbertragungssysteme. Aachen: Shaker Verlag."},{"issue":"9","key":"881_CR47","doi-asserted-by":"publisher","first-page":"3656","DOI":"10.1109\/TCSI.2021.3093172","volume":"68","author":"K Ochs","year":"2021","unstructured":"Ochs, K., Michaelis, D., & Jenderny, S. (2021). Synthesis of an equivalent circuit for spike-timing-dependent axon growth: What fires together now really wires together. IEEE Transactions on Circuits and Systems I: Regular Papers, 68(9), 3656\u20133667. https:\/\/doi.org\/10.1109\/TCSI.2021.3093172","journal-title":"IEEE Transactions on Circuits and Systems I: Regular Papers"},{"key":"881_CR48","doi-asserted-by":"crossref","unstructured":"O\u2019Connor, W. (2005). Wave and scattering methods for numerical simulation, Stefan Bilbao, Wiley, Chichester, UK, 2004, p. 380, \u00a3 75 (Hardback), ISBN: 0-470-87017-6: Book Reviews. Int J Numer Model. 18(4):325.","DOI":"10.1002\/jnm.565"},{"key":"881_CR49","unstructured":"Olsen, M. J., Werner, K. J., Germain, F. G. (2017). Network variable preserving step-size control in wave digital filters."},{"key":"881_CR50","doi-asserted-by":"crossref","unstructured":"Paul, A., Laurila, T., Vuorinen, V., & Divinski, S. (2014). In Fick\u2019s laws of diffusion; pp. 115\u2013139.","DOI":"10.1007\/978-3-319-07461-0_3"},{"issue":"11","key":"881_CR51","doi-asserted-by":"publisher","first-page":"1665","DOI":"10.1016\/0020-7225(92)90134-3","volume":"30","author":"MB Rubin","year":"1992","unstructured":"Rubin, M. B. (1992). Hyperbolic heat conduction and the second law. International Journal of Engineering Science, 30(11), 1665\u20131676. https:\/\/doi.org\/10.1016\/0020-7225(92)90134-3","journal-title":"International Journal of Engineering Science"},{"issue":"1582","key":"881_CR52","doi-asserted-by":"publisher","first-page":"144","DOI":"10.1016\/S1388-1981(02)00149-X","volume":"06","author":"C Sayas","year":"2002","unstructured":"Sayas, C., Avila, J., & Wandosell, F. (2002). Regulation of neuronal cytoskeleton by lysophosphatidic acid: Role of GSK-3. Biochimica et Biophysica Acta, 06(1582), 144\u201353.","journal-title":"Biochimica et Biophysica Acta"},{"key":"881_CR53","doi-asserted-by":"crossref","unstructured":"Singh Muralidhar, B. K., Ashkrizzadeh, R., Kohlstedt, H., Petraru, A. & Rieger R. (2022). A pressure-sensitive oscillator for neuromorphic applications. In 2022 IEEE biomedical circuits and systems conference (BioCAS); pp. 345\u2013348.","DOI":"10.1109\/BioCAS54905.2022.9948695"},{"key":"881_CR54","first-page":"473","volume":"98","author":"J Stefan","year":"1889","unstructured":"Stefan, J. (1889). \u00dcber einige probleme der theorie der w\u00e4rmeleitung. Sitzungber, Wien, Akad Mat Natur., 98, 473\u2013484.","journal-title":"Sitzungber, Wien, Akad Mat Natur."},{"issue":"1","key":"881_CR55","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1002\/aic.690130130","volume":"13","author":"LC Tao","year":"1967","unstructured":"Tao, L. C. (1967). Generalized numerical solutions of freezing a saturated liquid in cylinders and spheres. AIChE Journal, 13(1), 165\u2013169. https:\/\/doi.org\/10.1002\/aic.690130130","journal-title":"AIChE Journal"},{"issue":"2","key":"881_CR56","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1137\/0146018","volume":"46","author":"LN Tao","year":"1986","unstructured":"Tao, L. N. (1986). A method for solving moving boundary problems. SIAM Journal on Applied Mathematics, 46(2), 254\u2013264. https:\/\/doi.org\/10.1137\/0146018","journal-title":"SIAM Journal on Applied Mathematics"},{"issue":"05","key":"881_CR57","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/1913\/1\/012144","volume":"2021","author":"N Vaidya","year":"1913","unstructured":"Vaidya, N., Deshpande, A., & Pidurkar, S. (1913). Solution of heat equation (Partial Differential Equation) by various methods. Journal of Physics: Conference Series, 2021(05), 012144. https:\/\/doi.org\/10.1088\/1742-6596\/1913\/1\/012144","journal-title":"Journal of Physics: Conference Series"},{"key":"881_CR58","doi-asserted-by":"crossref","unstructured":"Vollmer, M. (2004). An approach to automatic generation of wave digital structures from PDEs. In 2004 IEEE International symposium on circuits and systems (IEEE Cat. No. 04CH37512), vol.\u00a03; pp. III\u2013245.","DOI":"10.1109\/ISCAS.2004.1328729"},{"issue":"4","key":"881_CR59","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/s11045-005-4125-4","volume":"16","author":"M Vollmer","year":"2005","unstructured":"Vollmer, M. (2005). Automatic generation of wave digital structures for numerically integrating linear symmetric hyperbolic PDEs. Multidimensional Systems and Signal Processing, 16(4), 369\u2013396. https:\/\/doi.org\/10.1007\/s11045-005-4125-4","journal-title":"Multidimensional Systems and Signal Processing"},{"issue":"19","key":"881_CR60","doi-asserted-by":"publisher","first-page":"6065","DOI":"10.1523\/JNEUROSCI.16-19-06065.1996","volume":"16","author":"J Wang","year":"1996","unstructured":"Wang, J., Yu, W., Baas, P. W., & Black, M. M. (1996). Microtubule assembly in growing dendrites. Journal of Neuroscience, 16(19), 6065\u20136078.","journal-title":"Journal of Neuroscience"},{"key":"881_CR61","doi-asserted-by":"crossref","unstructured":"Werner, K. J., Nangia, V., Smith, J. O. & Abel, J. S. (2015). A general and explicit formulation for wave digital filters with multiple\/multiport nonlinearities and complicated topologies. In 2015 IEEE workshop on applications of signal processing to audio and acoustics (WASPAA); pp. 1\u20135.","DOI":"10.1109\/WASPAA.2015.7336908"},{"issue":"343","key":"881_CR62","first-page":"407","volume":"01","author":"M Zerroukat","year":"1997","unstructured":"Zerroukat, M., & Chatwin, C. (1997). Computational moving boundary problem. Journal of Fluid Mechanics, 01(343), 407.","journal-title":"Journal of Fluid Mechanics"},{"key":"881_CR63","doi-asserted-by":"publisher","first-page":"9","DOI":"10.3389\/fenrg.2021.809434","volume":"12","author":"H Zhou","year":"2021","unstructured":"Zhou, H., Lu, T., Zhang, S., & Zhang, X. (2021). Lumped-circuits model of lossless transmission lines and its numerical characteristics. Frontiers in Energy Research., 12, 9. https:\/\/doi.org\/10.3389\/fenrg.2021.809434","journal-title":"Frontiers in Energy Research."}],"container-title":["Multidimensional Systems and Signal Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11045-023-00881-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11045-023-00881-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11045-023-00881-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,29]],"date-time":"2023-10-29T12:03:27Z","timestamp":1698581007000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11045-023-00881-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,24]]},"references-count":65,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["881"],"URL":"https:\/\/doi.org\/10.1007\/s11045-023-00881-z","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-2229820\/v1","asserted-by":"object"}]},"ISSN":["0923-6082","1573-0824"],"issn-type":[{"value":"0923-6082","type":"print"},{"value":"1573-0824","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,24]]},"assertion":[{"value":"2 November 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 July 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 July 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Financial or non-financial interests"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}