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approaches based on the methods of conditional and classical symrnetry reductions. In Chapter 1, we briefty introduce the equations that constitute magnetohy drodynamics. The fiow under consideration is assumed to be ideal, nonstationary and isentropic for a compressible conductive ftuid placed in a magnetic field J. The electrical conductivity of the fluid is assumed to be infinitely large. In Chapter 2, we present in Section 1 a version of the conditional symmetry method for resolving quasilinear hyperbolic systems of partial differential equations of the first order. The rank-one solutions (also called simple waves solutions) obtained by this method are expressed in terms of Riemann invariants. In Section 2, we use it to obtain simple waves of MHD system equations. Section 3 generalizes the above construction to the case of many simple waves described in terms of Riemann invariants. In Section 4 we present some double waves solutions admitted by the MHD equations. Finally, in Section 5 we impose some differential constraints in order to get some new classes of solutions. In Chapter 3, we have applied the symmetry reduction method to the MHD system of equations. The symmetry algebra and its classification by conju gacy classes of r-dimensional subalgebras (i r 4) was already known [17]. We restrict our study to the three dirnensional galilean similitude subalgebras that give systems of ordinary differential equation ftom which we obtain G-invariant solutions. 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Multiple Waves</a></div><div class="wp-workCard_item"><span>Journal of Nonlinear Mathematical Physics</span><span>, 2004</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0e2778c0639303a1cf3b77c0ea55f917" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":100095053,"asset_id":98856436,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/100095053/download_file?st=MTczMjc0NTEyNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="98856436"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="98856436"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 98856436; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=98856436]").text(description); $(".js-view-count[data-work-id=98856436]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 98856436; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='98856436']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 98856436, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0e2778c0639303a1cf3b77c0ea55f917" } } $('.js-work-strip[data-work-id=98856436]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":98856436,"title":"On Conditionally Invariant Solutions of Magnetohydrodynamic Equations. 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methods","url":"https://www.academia.edu/Documents/in/Exact_solution_methods"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="70236838"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/70236838/Reduction_and_Exact_Solutions_of_the_Ideal_Magnetohydrodynamic_Equations"><img alt="Research paper thumbnail of Reduction and Exact Solutions of the Ideal Magnetohydrodynamic Equations" class="work-thumbnail" src="https://attachments.academia-assets.com/80067534/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/70236838/Reduction_and_Exact_Solutions_of_the_Ideal_Magnetohydrodynamic_Equations">Reduction and Exact Solutions of the Ideal Magnetohydrodynamic Equations</a></div><div class="wp-workCard_item"><span>arXiv: Mathematical Physics</span><span>, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal mag...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal magnetohydrodynamic equations in (3+1) dimensions. These equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras ($1\leq r\leq 4$) was already known. So we restrict our study to the three-dimensional Galilean-similitude subalgebras that give systems composed of ordinary differential equations. We present here several examples of these solutions. Some of these exact solutions show interesting physical interpretations.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f8e7b0058d9b56318305506d80b4b313" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":80067534,"asset_id":70236838,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/80067534/download_file?st=MTczMjc0NTEyNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="70236838"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="70236838"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 70236838; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=70236838]").text(description); $(".js-view-count[data-work-id=70236838]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 70236838; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='70236838']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 70236838, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "f8e7b0058d9b56318305506d80b4b313" } } $('.js-work-strip[data-work-id=70236838]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":70236838,"title":"Reduction and Exact Solutions of the Ideal Magnetohydrodynamic Equations","translated_title":"","metadata":{"abstract":"In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal magnetohydrodynamic equations in (3+1) dimensions. 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In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1 r 4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. 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In Chapter 1, we briefty introduce the equations that constitute magnetohy drodynamics. The fiow under consideration is assumed to be ideal, nonstationary and isentropic for a compressible conductive ftuid placed in a magnetic field J. The electrical conductivity of the fluid is assumed to be infinitely large. In Chapter 2, we present in Section 1 a version of the conditional symmetry method for resolving quasilinear hyperbolic systems of partial differential equations of the first order. The rank-one solutions (also called simple waves solutions) obtained by this method are expressed in terms of Riemann invariants. In Section 2, we use it to obtain simple waves of MHD system equations. Section 3 generalizes the above construction to the case of many simple waves described in terms of Riemann invariants. In Section 4 we present some double waves solutions admitted by the MHD equations. Finally, in Section 5 we impose some differential constraints in order to get some new classes of solutions. In Chapter 3, we have applied the symmetry reduction method to the MHD system of equations. The symmetry algebra and its classification by conju gacy classes of r-dimensional subalgebras (i r 4) was already known [17]. We restrict our study to the three dirnensional galilean similitude subalgebras that give systems of ordinary differential equation ftom which we obtain G-invariant solutions. 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hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/98856436/On_Conditionally_Invariant_Solutions_of_Magnetohydrodynamic_Equations_Multiple_Waves"><img alt="Research paper thumbnail of On Conditionally Invariant Solutions of Magnetohydrodynamic Equations. Multiple Waves" class="work-thumbnail" src="https://attachments.academia-assets.com/100095053/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/98856436/On_Conditionally_Invariant_Solutions_of_Magnetohydrodynamic_Equations_Multiple_Waves">On Conditionally Invariant Solutions of Magnetohydrodynamic Equations. Multiple Waves</a></div><div class="wp-workCard_item"><span>Journal of Nonlinear Mathematical Physics</span><span>, 2004</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0e2778c0639303a1cf3b77c0ea55f917" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":100095053,"asset_id":98856436,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/100095053/download_file?st=MTczMjc0NTEyNiw4LjIyMi4yMDguMTQ2&st=MTczMjc0NTEyNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="98856436"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="98856436"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 98856436; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=98856436]").text(description); $(".js-view-count[data-work-id=98856436]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 98856436; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='98856436']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 98856436, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0e2778c0639303a1cf3b77c0ea55f917" } } $('.js-work-strip[data-work-id=98856436]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":98856436,"title":"On Conditionally Invariant Solutions of Magnetohydrodynamic Equations. Multiple Waves","translated_title":"","metadata":{"publisher":"Informa UK Limited","grobid_abstract":"We present a version of the conditional symmetry method in order to obtain multiple wave solutions expressed in terms of Riemann invariants. We construct an abelian distribution of vector fields which are symmetries of the original system of PDEs subjected to certain first order differential constraints. The usefulness of our approach is demonstrated on simple and double wave solutions of MHD equations. 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href="https://www.academia.edu/70236838/Reduction_and_Exact_Solutions_of_the_Ideal_Magnetohydrodynamic_Equations">Reduction and Exact Solutions of the Ideal Magnetohydrodynamic Equations</a></div><div class="wp-workCard_item"><span>arXiv: Mathematical Physics</span><span>, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal mag...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal magnetohydrodynamic equations in (3+1) dimensions. These equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras ($1\leq r\leq 4$) was already known. So we restrict our study to the three-dimensional Galilean-similitude subalgebras that give systems composed of ordinary differential equations. We present here several examples of these solutions. Some of these exact solutions show interesting physical interpretations.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f8e7b0058d9b56318305506d80b4b313" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":80067534,"asset_id":70236838,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/80067534/download_file?st=MTczMjc0NTEyNiw4LjIyMi4yMDguMTQ2&st=MTczMjc0NTEyNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="70236838"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="70236838"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 70236838; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=70236838]").text(description); $(".js-view-count[data-work-id=70236838]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 70236838; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='70236838']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 70236838, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "f8e7b0058d9b56318305506d80b4b313" } } $('.js-work-strip[data-work-id=70236838]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":70236838,"title":"Reduction and Exact Solutions of the Ideal Magnetohydrodynamic Equations","translated_title":"","metadata":{"abstract":"In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal magnetohydrodynamic equations in (3+1) dimensions. These equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras ($1\\leq r\\leq 4$) was already known. So we restrict our study to the three-dimensional Galilean-similitude subalgebras that give systems composed of ordinary differential equations. We present here several examples of these solutions. Some of these exact solutions show interesting physical interpretations.","publication_date":{"day":null,"month":null,"year":2005,"errors":{}},"publication_name":"arXiv: Mathematical Physics"},"translated_abstract":"In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal magnetohydrodynamic equations in (3+1) dimensions. These equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras ($1\\leq r\\leq 4$) was already known. 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