Pub Date : 2023-05-08DOI: 10.1017/S0956792523000013
Barbara A. Wagner, M. Timme
The conversion, storage and distribution of energy from renewable sources drive some of the most inno-vative technologies with the goal to meet global energy demands and to mitigate climate change. For instance, research in photovoltaics, which combines materials science, device modelling and optics, has seen intensive growth in all areas and on all relevant scales during the past decades. The role of mathematics has been key to understand and develop novel optimised photovoltaic devices. However, while the global cumulative solar capacity is growing fast, the temporal variability of photovoltaic or wind-generated electricity from intra-day to seasonal scales constitutes a major obstacle for matching demand. Energy storage plays a major role in addressing this problem, for example via large-scale stationary battery systems, or photochemical hydrogen production. Most importantly, the growing complexity of power distribution across coupled distribution grids constitutes perhaps the hardest current challenge. Apart from setting up the infrastructure, monitoring and planning, these increasingly complex networks pose difficult mathematical problems, relating to fluctuations in coupled energy networks, that are also impacted by market regulations. The focus of research articles in this special issue of the European Journal of Applied Mathematics presents some of the mathematical challenges encountered in the prediction of power grid responses on different temporal and spatial scales and to various types of disturbances. In particular, contributions to the special issue report findings on coupled energy networks composed of gas and electric power networks of realistic size, that are coupled to stochastic fluctuations due to fluctuating demands and supplies. Others address the stability of electrical power grids, develop guiding principles for power grid operation, control and design, and perform case studies on optimisation of energy generation, taking into account sustainability goals for 2050 in Europe, under uncertain future market conditions. In addition, a modelling framework that enables researchers to quantify degradation effects of lithium-ion battery cells and a new approach to model parabolic trough power plants are presented. Such complex problems require a broad spectrum of
{"title":"Special Issue of EJAM: The Mathematics in Renewable Energy","authors":"Barbara A. Wagner, M. Timme","doi":"10.1017/S0956792523000013","DOIUrl":"https://doi.org/10.1017/S0956792523000013","url":null,"abstract":"The conversion, storage and distribution of energy from renewable sources drive some of the most inno-vative technologies with the goal to meet global energy demands and to mitigate climate change. For instance, research in photovoltaics, which combines materials science, device modelling and optics, has seen intensive growth in all areas and on all relevant scales during the past decades. The role of mathematics has been key to understand and develop novel optimised photovoltaic devices. However, while the global cumulative solar capacity is growing fast, the temporal variability of photovoltaic or wind-generated electricity from intra-day to seasonal scales constitutes a major obstacle for matching demand. Energy storage plays a major role in addressing this problem, for example via large-scale stationary battery systems, or photochemical hydrogen production. Most importantly, the growing complexity of power distribution across coupled distribution grids constitutes perhaps the hardest current challenge. Apart from setting up the infrastructure, monitoring and planning, these increasingly complex networks pose difficult mathematical problems, relating to fluctuations in coupled energy networks, that are also impacted by market regulations. The focus of research articles in this special issue of the European Journal of Applied Mathematics presents some of the mathematical challenges encountered in the prediction of power grid responses on different temporal and spatial scales and to various types of disturbances. In particular, contributions to the special issue report findings on coupled energy networks composed of gas and electric power networks of realistic size, that are coupled to stochastic fluctuations due to fluctuating demands and supplies. Others address the stability of electrical power grids, develop guiding principles for power grid operation, control and design, and perform case studies on optimisation of energy generation, taking into account sustainability goals for 2050 in Europe, under uncertain future market conditions. In addition, a modelling framework that enables researchers to quantify degradation effects of lithium-ion battery cells and a new approach to model parabolic trough power plants are presented. Such complex problems require a broad spectrum of","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":"34 1","pages":"425 - 428"},"PeriodicalIF":1.9,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44704596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-24DOI: 10.1017/s0956792523000104
Novrianti, O. Sawada, N. Tsuge
The time-global unique classical positive solutions to the reaction–diffusion equations for prey–predator models with dormancy of predators are constructed. The feature appears on the nonlinear terms of Holling type $rm I!I$ functional response. The crucial step is to establish time-local positive classical solutions by using a new approximation associated with time-evolution operators. Although the system does not equip usual comparison principle for solutions to partial differential equation, a priori bounds are derived by enclosing and renormalising arguments of solutions to the corresponding ordinary differential equations. Furthermore, time-global existence, invariant regions and asymptotic behaviours of solutions follow from such a priori bounds.
{"title":"Positive solutions to the prey–predator equations with dormancy of predators","authors":"Novrianti, O. Sawada, N. Tsuge","doi":"10.1017/s0956792523000104","DOIUrl":"https://doi.org/10.1017/s0956792523000104","url":null,"abstract":"\u0000 The time-global unique classical positive solutions to the reaction–diffusion equations for prey–predator models with dormancy of predators are constructed. The feature appears on the nonlinear terms of Holling type \u0000 \u0000 \u0000 \u0000$rm I!I$\u0000\u0000 \u0000 functional response. The crucial step is to establish time-local positive classical solutions by using a new approximation associated with time-evolution operators. Although the system does not equip usual comparison principle for solutions to partial differential equation, a priori bounds are derived by enclosing and renormalising arguments of solutions to the corresponding ordinary differential equations. Furthermore, time-global existence, invariant regions and asymptotic behaviours of solutions follow from such a priori bounds.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45719203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-20DOI: 10.1017/s0956792523000116
Ganghua Yuan, Yue Zhao
This paper is concerned with the increasing stability of the inverse source problem for the elastic wave equation with attenuation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the attenuation coefficient. The ingredients of the analysis include Carleman estimates and time decay estimates for the elastic wave equation to obtain an exact observability bound, and the study of the resonance-free region and an upper bound of the resolvent in this region for the elliptic operator with respect to the complex frequency. The advantage of the method developed in this work is that it can be used to study the case of variable attenuation coefficient.
{"title":"Increasing stability for the inverse source problem in elastic waves with attenuation","authors":"Ganghua Yuan, Yue Zhao","doi":"10.1017/s0956792523000116","DOIUrl":"https://doi.org/10.1017/s0956792523000116","url":null,"abstract":"\u0000 This paper is concerned with the increasing stability of the inverse source problem for the elastic wave equation with attenuation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the attenuation coefficient. The ingredients of the analysis include Carleman estimates and time decay estimates for the elastic wave equation to obtain an exact observability bound, and the study of the resonance-free region and an upper bound of the resolvent in this region for the elliptic operator with respect to the complex frequency. The advantage of the method developed in this work is that it can be used to study the case of variable attenuation coefficient.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45282015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-23DOI: 10.1017/s0956792523000050
M. Assunção, M. Vynnycky
The one-dimensional transient solidification of a binary alloy undergoing shrinkage is well-known as an invaluable benchmark for the testing of numerical codes that model macrosegregation. Here, recent work that considered the small-time behaviour of this problem is extended until complete solidification, thereby determining the solute profile across the entire solidified domain. The small-time solution is used as the initial condition for the numerical integration of a problem having three moving boundaries. Of particular significance is the so-called inverse segregation that is observed at the start of solidification, and the extreme segregation that is observed at the end; in the case of the example presented, which is for the often-cited Al–Cu system, the macrosegregation is found to be positive or negative, depending on whether Scheil’s equation or the lever rule is assumed at the microscale, respectively. The relevance of these results for the modelling of steady-state continuous casting processes – in particular, the phenomenon of centreline segregation – is also discussed.
{"title":"On macrosegregation in a binary alloy undergoing solidification shrinkage","authors":"M. Assunção, M. Vynnycky","doi":"10.1017/s0956792523000050","DOIUrl":"https://doi.org/10.1017/s0956792523000050","url":null,"abstract":"\u0000 The one-dimensional transient solidification of a binary alloy undergoing shrinkage is well-known as an invaluable benchmark for the testing of numerical codes that model macrosegregation. Here, recent work that considered the small-time behaviour of this problem is extended until complete solidification, thereby determining the solute profile across the entire solidified domain. The small-time solution is used as the initial condition for the numerical integration of a problem having three moving boundaries. Of particular significance is the so-called inverse segregation that is observed at the start of solidification, and the extreme segregation that is observed at the end; in the case of the example presented, which is for the often-cited Al–Cu system, the macrosegregation is found to be positive or negative, depending on whether Scheil’s equation or the lever rule is assumed at the microscale, respectively. The relevance of these results for the modelling of steady-state continuous casting processes – in particular, the phenomenon of centreline segregation – is also discussed.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49556171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-02DOI: 10.1017/s0956792523000025
M. El Jarroudi, Mhamed El Merzguioui, M. Er-Riani, A. Lahrouz, Jamal El Amrani
The aim of this paper is to study the dimension reduction analysis of an elastic plate with small thickness reinforced with increasing number of thin ribbons developing fractal geometry. We prove the $Gamma $ -convergence of the energy functionals to a two-dimensional effective energy including singular terms supported within the Sierpinski carpet.
{"title":"Dimension reduction analysis of a three-dimensional thin elastic plate reinforced with fractal ribbons","authors":"M. El Jarroudi, Mhamed El Merzguioui, M. Er-Riani, A. Lahrouz, Jamal El Amrani","doi":"10.1017/s0956792523000025","DOIUrl":"https://doi.org/10.1017/s0956792523000025","url":null,"abstract":"\u0000 The aim of this paper is to study the dimension reduction analysis of an elastic plate with small thickness reinforced with increasing number of thin ribbons developing fractal geometry. We prove the \u0000 \u0000 \u0000 \u0000$Gamma $\u0000\u0000 \u0000 -convergence of the energy functionals to a two-dimensional effective energy including singular terms supported within the Sierpinski carpet.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41440314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-03DOI: 10.1017/s0956792523000049
D. Huang, Shanshan Chen
In this paper, we consider a general single population model with delay and patch structure, which could model the population loss during the dispersal. It is shown that the model admits a unique positive equilibrium when the dispersal rate is smaller than a critical value. The stability of the positive equilibrium and associated Hopf bifurcation are investigated when the dispersal rate is small or near the critical value. Moreover, we show the effect of network topology on Hopf bifurcation values for a delayed logistic population model.
{"title":"Dynamics of a delayed population patch model with the dispersion matrix incorporating population loss","authors":"D. Huang, Shanshan Chen","doi":"10.1017/s0956792523000049","DOIUrl":"https://doi.org/10.1017/s0956792523000049","url":null,"abstract":"\u0000 In this paper, we consider a general single population model with delay and patch structure, which could model the population loss during the dispersal. It is shown that the model admits a unique positive equilibrium when the dispersal rate is smaller than a critical value. The stability of the positive equilibrium and associated Hopf bifurcation are investigated when the dispersal rate is small or near the critical value. Moreover, we show the effect of network topology on Hopf bifurcation values for a delayed logistic population model.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43247560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-01DOI: 10.1017/S0956792522000389
M. Gugat, J. Habermann, M. Hintermüller, Olivier Huber
Abstract While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.
{"title":"Constrained exact boundary controllability of a semilinear model for pipeline gas flow","authors":"M. Gugat, J. Habermann, M. Hintermüller, Olivier Huber","doi":"10.1017/S0956792522000389","DOIUrl":"https://doi.org/10.1017/S0956792522000389","url":null,"abstract":"Abstract While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":"34 1","pages":"532 - 553"},"PeriodicalIF":1.9,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47476058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-23DOI: 10.1017/s0956792522000419
E. Luckins, C. Breward, I. Griffiths, C. Please
Evaporation within porous media is both a multiscale and interface-driven process, since the phase change at the evaporating interfaces within the pores generates a vapour flow and depends on the transport of vapour through the porous medium. While homogenised models of flow and chemical transport in porous media allow multiscale processes to be modelled efficiently, it is not clear how the multiscale effects impact the interface conditions required for these homogenised models. In this paper, we derive a homogenised model, including effective interface conditions, for the motion of an evaporation front through a porous medium, using a combined homogenisation and boundary layer analysis. This analysis extends previous work for a purely diffusive problem to include both gas flow and the advective–diffusive transport of material. We investigate the effect that different microscale models describing the chemistry of the evaporation have on the homogenised interface conditions. In particular, we identify a new effective parameter, $mathcal{L}$ , the average microscale interface length, which modifies the effective evaporation rate in the homogenised model. Like the effective diffusivity and permeability of a porous medium, $mathcal{L}$ may be found by solving a periodic cell problem on the microscale. We also show that the different microscale models of the interface chemistry result in fundamentally different fine-scale behaviour at, and near, the interface.
{"title":"A homogenised model for the motion of evaporating fronts in porous media","authors":"E. Luckins, C. Breward, I. Griffiths, C. Please","doi":"10.1017/s0956792522000419","DOIUrl":"https://doi.org/10.1017/s0956792522000419","url":null,"abstract":"\u0000 Evaporation within porous media is both a multiscale and interface-driven process, since the phase change at the evaporating interfaces within the pores generates a vapour flow and depends on the transport of vapour through the porous medium. While homogenised models of flow and chemical transport in porous media allow multiscale processes to be modelled efficiently, it is not clear how the multiscale effects impact the interface conditions required for these homogenised models. In this paper, we derive a homogenised model, including effective interface conditions, for the motion of an evaporation front through a porous medium, using a combined homogenisation and boundary layer analysis. This analysis extends previous work for a purely diffusive problem to include both gas flow and the advective–diffusive transport of material. We investigate the effect that different microscale models describing the chemistry of the evaporation have on the homogenised interface conditions. In particular, we identify a new effective parameter, \u0000 \u0000 \u0000 \u0000$mathcal{L}$\u0000\u0000 \u0000 , the average microscale interface length, which modifies the effective evaporation rate in the homogenised model. Like the effective diffusivity and permeability of a porous medium, \u0000 \u0000 \u0000 \u0000$mathcal{L}$\u0000\u0000 \u0000 may be found by solving a periodic cell problem on the microscale. We also show that the different microscale models of the interface chemistry result in fundamentally different fine-scale behaviour at, and near, the interface.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47679907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-16DOI: 10.1017/s0956792522000377
W. Casper, Taylor Grimes
The classic game of Battleship involves two players taking turns attempting to guess the positions of a fleet of vertically or horizontally positioned enemy ships hidden on a $10times 10$ grid. One variant of this game, also referred to as Battleship Solitaire, Bimaru or Yubotu, considers the game with the inclusion of X-ray data, represented by knowledge of how many spots are occupied in each row and column in the enemy board. This paper considers the Battleship puzzle problem: the problem of reconstructing an enemy fleet from its X-ray data. We generate non-unique solutions to Battleship puzzles via certain reflection transformations akin to Ryser interchanges. Furthermore, we demonstrate that solutions of Battleship puzzles may be reliably obtained by searching for solutions of the associated classical binary discrete tomography problem which minimise the discrete Laplacian. We reformulate this optimisation problem as a quadratic unconstrained binary optimisation problem and approximate solutions via a simulated annealer, emphasising the future practical applicability of quantum annealers to solving discrete tomography problems with predefined structure.
{"title":"Battleship, tomography and quantum annealing","authors":"W. Casper, Taylor Grimes","doi":"10.1017/s0956792522000377","DOIUrl":"https://doi.org/10.1017/s0956792522000377","url":null,"abstract":"\u0000 The classic game of Battleship involves two players taking turns attempting to guess the positions of a fleet of vertically or horizontally positioned enemy ships hidden on a \u0000 \u0000 \u0000 \u0000$10times 10$\u0000\u0000 \u0000 grid. One variant of this game, also referred to as Battleship Solitaire, Bimaru or Yubotu, considers the game with the inclusion of X-ray data, represented by knowledge of how many spots are occupied in each row and column in the enemy board. This paper considers the Battleship puzzle problem: the problem of reconstructing an enemy fleet from its X-ray data. We generate non-unique solutions to Battleship puzzles via certain reflection transformations akin to Ryser interchanges. Furthermore, we demonstrate that solutions of Battleship puzzles may be reliably obtained by searching for solutions of the associated classical binary discrete tomography problem which minimise the discrete Laplacian. We reformulate this optimisation problem as a quadratic unconstrained binary optimisation problem and approximate solutions via a simulated annealer, emphasising the future practical applicability of quantum annealers to solving discrete tomography problems with predefined structure.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43618966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-16DOI: 10.1017/S0956792523000037
P. Hydon
Abstract The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.
{"title":"Partial Euler operators and the efficient inversion of Div","authors":"P. Hydon","doi":"10.1017/S0956792523000037","DOIUrl":"https://doi.org/10.1017/S0956792523000037","url":null,"abstract":"Abstract The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":"34 1","pages":"1046 - 1066"},"PeriodicalIF":1.9,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48745043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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