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<title>Boundary value problem</title>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#ccccff;display:block;margin-bottom:0.2em;"><a href="Differential_equation" title="Differential equation">Differential equations</a></th></tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;">

Scope</th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Fields</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="padding-bottom:0;">

<div class="hlist"><ul><li><a href="Natural_science" title="Natural science">Natural sciences</a></li><li><a href="Engineering" title="Engineering">Engineering</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;">

<ul><li><a href="Astronomy" title="Astronomy">Astronomy</a></li>

<li><a href="Physics" title="Physics">Physics</a></li>

<li><a href="Chemistry" title="Chemistry">Chemistry</a></li>

<li><br><a href="Biology" title="Biology">Biology</a></li>

<li><a href="Geology" title="Geology">Geology</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="padding-bottom:0;">

<a href="Applied_mathematics" title="Applied mathematics">Applied mathematics</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;">

<ul><li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li>

<li><a href="Chaos_theory" title="Chaos theory">Chaos theory</a></li>

<li><a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="padding-bottom:0;">

<a href="Social_science" title="Social science">Social sciences</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;;padding-bottom:0;">

<ul><li><a href="Economics" title="Economics">Economics</a></li>

<li><a href="Population_dynamics" title="Population dynamics">Population dynamics</a></li></ul></td>

</tr></tbody></table>

<hr>

<a href="List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></div></div></td>

</tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;;display:block;margin-top:0.1em;">

Classification</th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Types</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content">

<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">Partial</a></li>

<li><a href="Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li>

<li><a href="Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li>

<li><a href="Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li>

<li><a href="Linear_differential_equation" title="Linear differential equation">Linear</a></li>

<li><a href="Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li></ul>

</div></td>

</tr><tr><th class="sidebar-heading">

By variable type</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li></ul>

<ul><li><a href="Autonomous_differential_equation" class="mw-redirect" title="Autonomous differential equation">Autonomous</a></li>

<li>Coupled / Decoupled</li>

<li><a href="Exact_differential_equation" title="Exact differential equation">Exact</a></li>

<li><a href="Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a> / <a href="Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li></ul>

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Features</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Ordinary_differential_equation#Definitions" title="Ordinary differential equation">Order</a></li>

<li><a href="Differential_operator" title="Differential operator">Operator</a></li></ul>

</div>

<ul><li><a href="Notation_for_differentiation" title="Notation for differentiation">Notation</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Relation to processes</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;">

<ul><li><a href="Difference_equation" class="mw-redirect" title="Difference equation">Difference <span style="font-size: 85%;">(discrete analogue)</span></a></li></ul>

<ul><li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a>

<ul><li><a href="Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li>

<li><a href="Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul>

</div></div></div></td>

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Solution</th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Existence and uniqueness</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;">

<ul><li><a href="Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem </a></li>

<li><a href="Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li>

<li><a href="Carath%C3%A9odory's_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li>

<li><a href="Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">General topics</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist">

<ul><li><a href="Initial_condition" title="Initial condition">Initial conditions</a></li>

<li>

<ul><li><a href="Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet</a></li>

<li><a href="Neumann_boundary_condition" title="Neumann boundary condition">Neumann</a></li>

<li><a href="Robin_boundary_condition" title="Robin boundary condition">Robin</a></li>

<li><a href="Cauchy_problem" title="Cauchy problem">Cauchy problem</a></li></ul></li>

<li><a href="Wronskian" title="Wronskian">Wronskian</a></li>

<li><a href="Phase_portrait" title="Phase portrait">Phase portrait</a></li>

<li><a href="Lyapunov_stability" title="Lyapunov stability">Lyapunov</a> / <a href="Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic</a> / <a href="Exponential_stability" title="Exponential stability">Exponential stability</a></li>

<li><a href="Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li>

<li><span class="nowrap"><a href="Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series</a> / Integral solutions</span></li>

<li><a href="Numerical_integration" title="Numerical integration">Numerical integration</a></li>

<li><a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Solution methods</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist">

<ul><li>Inspection</li>

<li><a href="Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li>

<li><br><a href="Euler_method" title="Euler method">Euler</a></li>

<li><a href="Exponential_response_formula" title="Exponential response formula">Exponential response formula</a></li>

<li><a href="Finite_difference_method" title="Finite difference method">Finite difference</a> <span style="font-size: 85%;">(<a href="Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a>)</span></li>

<li><a href="Finite_element_method" title="Finite element method">Finite element</a>

<ul><li><a href="Infinite_element_method" title="Infinite element method">Infinite element</a></li></ul></li>

<li><a href="Finite_volume_method" title="Finite volume method">Finite volume</a></li>

<li><a href="Galerkin_method" title="Galerkin method">Galerkin</a>

<ul><li><a href="Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin</a></li></ul></li>

<li><a href="Green's_function" title="Green's function">Green's function</a></li>

<li><a href="Integrating_factor" title="Integrating factor">Integrating factor</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transforms</a></li>

<li><a href="Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li>

<li><a href="Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta</a></li></ul>

</div>

<ul><li><a href="Separation_of_variables" title="Separation of variables">Separation of variables</a></li>

<li><a href="Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Undetermined coefficients</a></li>

<li><a href="Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li></ul></div></div></td>

</tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;">

People</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li>

<li><a href="Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a></li>

<li><a href="Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li>

<li><a href="J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li>

<li><a href="Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="George_Green_(mathematician)" title="George Green (mathematician)">George Green</a></li>

<li><a href="Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li>

<li><a href="Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li>

<li><a href="Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li>

<li><a href="Ernst_Lindel%C3%B6f" class="mw-redirect" title="Ernst Lindelöf">Ernst Lindelöf</a></li>

<li><a href="%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li>

<li><a href="Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li>

<li><a href="John_Crank" title="John Crank">John Crank</a></li></ul>

</div></div></div></td>

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<p>In the study of <a href="Differential_equation" title="Differential equation">differential equations</a>, a <b>boundary-value problem</b> is a <a href="Differential_equation" title="Differential equation">differential equation</a> subjected to constraints called <b>boundary conditions</b>.<sup id="cite_ref-Zwillinger2014_1-0" class="reference"><a href="#cite_note-Zwillinger2014-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

</p><p>Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the <a href="Wave_equation" title="Wave equation">wave equation</a>, such as the determination of <a href="Normal_mode" title="Normal mode">normal modes</a>, are often stated as boundary value problems. A large class of important boundary value problems are the <a href="Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville problems</a>. The analysis of these problems, in the linear case, involves the <a href="Eigenfunction" title="Eigenfunction">eigenfunctions</a> of a <a href="Differential_operator" title="Differential operator">differential operator</a>.

</p><p>To be useful in applications, a boundary value problem should be <a href="Well-posed_problem" title="Well-posed problem">well posed</a>. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a> is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.

</p><p>Among the earliest boundary value problems to be studied is the <a href="Dirichlet_problem" title="Dirichlet problem">Dirichlet problem</a>, of finding the <a href="Harmonic_function" title="Harmonic function">harmonic functions</a> (solutions to <a href="Laplace's_equation" title="Laplace's equation">Laplace's equation</a>); the solution was given by the <a href="Dirichlet's_principle" title="Dirichlet's principle">Dirichlet's principle</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Explanation">Explanation</h2></div>

<p>Boundary value problems are similar to <a href="Initial_value_problem" title="Initial value problem">initial value problems</a>. A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value). A <b>boundary value</b> is a data value that corresponds to a minimum or maximum input, internal, or output value specified for a system or component.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>For example, if the independent variable is time over the domain [0,1], a boundary value problem would specify values for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}">

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<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

</semantics>

</math></span><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=1}</annotation>

</semantics>

</math></span><img src="./970dea4a5f5ec5355c4cdd62f6396fbc8b1baaa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=1}" loading="lazy"></span>, whereas an initial value problem would specify a value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation>

</semantics>

</math></span><img src="./397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'(t)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y'(t)}</annotation>

</semantics>

</math></span><img src="./ac415aa71b96af9b4e78aea31eff4ba122383095.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.494ex; height:3.009ex;" alt="{\displaystyle y'(t)}" loading="lazy"></span> at time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

</semantics>

</p><p>Finding the temperature at all points of an iron bar with one end kept at <a href="Absolute_zero" title="Absolute zero">absolute zero</a> and the other end at the freezing point of water would be a boundary value problem.

</p><p>If the problem is dependent on both space and time, one could specify the value of the problem at a given point for all time or at a given time for all space.

</p><p>Concretely, an example of a boundary value problem (in one spatial dimension) is

</p>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y''(x)+y(x)=0}</annotation>

</semantics>

</math></span><img src="./6ffb5c77128a9bcaf53a4a87bdfcc1b45739fa33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.832ex; height:3.009ex;" alt="{\displaystyle y''(x)+y(x)=0}" loading="lazy"></span></dd></dl>

<p>to be solved for the unknown function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(x)}</annotation>

</semantics>

</math></span><img src="./e871993bfd131a8b0c3591c26084cf8171a74dcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.294ex; height:2.843ex;" alt="{\displaystyle y(x)}" loading="lazy"></span> with the boundary conditions

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(0)=0,\ y(\pi /2)=2.}</annotation>

</semantics>

</math></span><img src="./1c34653e3fd96f0d46c39795d33c8f1bf4f24a1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.532ex; height:2.843ex;" alt="{\displaystyle y(0)=0,\ y(\pi /2)=2.}" loading="lazy"></span></dd></dl>

<p>Without the boundary conditions, the general solution to this equation is

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(x)=A\sin(x)+B\cos(x).}</annotation>

</semantics>

</math></span><img src="./cb4529e2702bacf7586073389c2be2f9f9cd67fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.406ex; height:2.843ex;" alt="{\displaystyle y(x)=A\sin(x)+B\cos(x).}" loading="lazy"></span></dd></dl>

<p>From the boundary condition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(0)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(0)=0}</annotation>

</semantics>

</math></span><img src="./343c32f38bb379b4b208477b130d8f522d3f0788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.388ex; height:2.843ex;" alt="{\displaystyle y(0)=0}" loading="lazy"></span> one obtains

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0=A\cdot 0+B\cdot 1}</annotation>

</semantics>

</math></span><img src="./c2807e83dcb0a845448ccb47588ff9c894ec90e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.292ex; height:2.343ex;" alt="{\displaystyle 0=A\cdot 0+B\cdot 1}" loading="lazy"></span></dd></dl>

<p>which implies that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B=0.}</annotation>

</semantics>

</math></span><img src="./c167bcbf56ca94e6de4d11644b1de18345c66bea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.672ex; height:2.176ex;" alt="{\displaystyle B=0.}" loading="lazy"></span> From the boundary condition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(\pi /2)=2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(\pi /2)=2}</annotation>

</semantics>

</math></span><img src="./f600538e5b49a27b2227273eefe58d7d0f0bbedf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.883ex; height:2.843ex;" alt="{\displaystyle y(\pi /2)=2}" loading="lazy"></span> one finds

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2=A\cdot 1}</annotation>

</semantics>

</math></span><img src="./eab14817486bc2d6409ed9956a69d323b0530fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.846ex; height:2.176ex;" alt="{\displaystyle 2=A\cdot 1}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=2.}</annotation>

</semantics>

</math></span><img src="./264d00c42d4674ebb21b9fd723bb60123fc6ea0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.651ex; height:2.176ex;" alt="{\displaystyle A=2.}" loading="lazy"></span> One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(x)=2\sin(x).}</annotation>

</semantics>

</math></span><img src="./1e7f068aeceb272eaa7371ab981fc264b9b5a023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.584ex; height:2.843ex;" alt="{\displaystyle y(x)=2\sin(x).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Types_of_boundary_value_problems">Types of boundary value problems</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Boundary_value_conditions">Boundary value conditions</h3></div>

<p>A boundary condition which specifies the value of the function itself is a <a href="Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet boundary condition</a>, or first-type boundary condition. For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space.

</p><p>A boundary condition which specifies the value of the <a href="Normal_derivative" class="mw-redirect" title="Normal derivative">normal derivative</a> of the function is a <a href="Neumann_boundary_condition" title="Neumann boundary condition">Neumann boundary condition</a>, or second-type boundary condition. For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known.

</p><p>If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a <a href="Cauchy_boundary_condition" title="Cauchy boundary condition">Cauchy boundary condition</a>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4></div>

<p>Summary of boundary conditions for the unknown function, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>, constants <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math></span><img src="./1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1}}</annotation>

</semantics>

</math></span><img src="./77b7dc6d279091d354e0b90889b463bfa7eb7247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{1}}" loading="lazy"></span> specified by the boundary conditions, and known scalar functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> specified by the boundary conditions.

</p>

<th>Name

</th>

<th>Form on 1st part of boundary

</th>

<th>Form on 2nd part of boundary

</th></tr>

<tr>

<td><a href="Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet</a>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=f}</annotation>

</semantics>

</td></tr>

<tr>

<td><a href="Neumann_boundary_condition" title="Neumann boundary condition">Neumann</a>

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\partial y \over \partial n}=f}</annotation>

</semantics>

</math></span><img src="./5533bf993cc5dc02dd2536287b81f89e92c3aa78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.926ex; height:5.676ex;" alt="{\displaystyle {\partial y \over \partial n}=f}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Robin_boundary_condition" title="Robin boundary condition">Robin</a>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}y+c_{1}{\partial y \over \partial n}=f}</annotation>

</semantics>

</math></span><img src="./c4d4d5514cd01646b43e3b540eb4e7338f89d315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.044ex; height:5.676ex;" alt="{\displaystyle c_{0}y+c_{1}{\partial y \over \partial n}=f}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Mixed_boundary_condition" title="Mixed boundary condition">Mixed</a>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=f}</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{0}y+c_{1}{\partial y \over \partial n}=g}</annotation>

</semantics>

</math></span><img src="./b4d2c35a2faaa006bcef3652ecb994cdded3b02d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.881ex; height:5.676ex;" alt="{\displaystyle c_{0}y+c_{1}{\partial y \over \partial n}=g}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Cauchy_boundary_condition" title="Cauchy boundary condition">Cauchy</a>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=f}</annotation>

</semantics>

</math></span><img src="./2db59eb95797f764fd98e198d6579d322bcfd61d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.533ex; height:2.509ex;" alt="{\displaystyle y=f}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial y \over \partial n}=g}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\partial y \over \partial n}=g}</annotation>

</semantics>

</math></span><img src="./6929e0454a5b693aedb3f0fa15370fee78efdb99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.763ex; height:5.676ex;" alt="{\displaystyle {\partial y \over \partial n}=g}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Differential_operators">Differential operators</h3></div>

<p>Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an <a href="Elliptic_operator" title="Elliptic operator">elliptic operator</a>, one discusses <a href="Elliptic_boundary_value_problem" title="Elliptic boundary value problem">elliptic boundary value problems</a>. For a <a href="Hyperbolic_operator" class="mw-redirect" title="Hyperbolic operator">hyperbolic operator</a>, one discusses hyperbolic boundary value problems. These categories are further subdivided into <a href="Linear_differential_equation" title="Linear differential equation">linear</a> and various nonlinear types.

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Electromagnetic_potential">Electromagnetic potential</h3></div>

/* start https://en.wikipedia.org/ */

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Laplace's_equation#Boundary_conditions" title="Laplace's equation">Laplace's equation § Boundary conditions</a></div>

<p>In <a href="Electrostatics" title="Electrostatics">electrostatics</a>, a common problem is to find a function which describes the <a href="Electric_potential" title="Electric potential">electric potential</a> of a given region. If the region does not contain charge, the potential must be a solution to <a href="Laplace's_equation" title="Laplace's equation">Laplace's equation</a> (a so-called <a href="Harmonic_function" title="Harmonic function">harmonic function</a>). The boundary conditions in this case are the <a href="Interface_conditions_for_electromagnetic_fields" title="Interface conditions for electromagnetic fields">Interface conditions for electromagnetic fields</a>. If there is no <a href="Current_density" title="Current density">current density</a> in the region, it is also possible to define a <a href="Magnetic_scalar_potential" title="Magnetic scalar potential">magnetic scalar potential</a> using a similar procedure.

</p>

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</style><div>

<p><b>Related mathematics:</b>

</p>

<ul><li><a href="Initial_value_problem" title="Initial value problem">Initial value problem</a></li>

<li><a href="Green's_function" title="Green's function">Green's function</a></li>

<li><a href="Stochastic_processes_and_boundary_value_problems" title="Stochastic processes and boundary value problems">Stochastic processes and boundary value problems</a></li>

<li><a href="Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a></li>

<li><a href="Sommerfeld_radiation_condition" title="Sommerfeld radiation condition">Sommerfeld radiation condition</a></li>

<li><a href="Perfect_thermal_contact" title="Perfect thermal contact">Perfect thermal contact condition</a></li></ul>

</td>

<p><b>Physical applications:</b>

</p>

<ul><li><a href="Wave" title="Wave">Waves</a></li>

<li><a href="Normal_mode" title="Normal mode">Normal mode</a></li>

<li><a href="Electrostatics" title="Electrostatics">Electrostatics</a></li>

<li><a href="Potential_theory" title="Potential theory">Potential theory</a></li>

<li><a href="Computation_of_radiowave_attenuation_in_the_atmosphere" class="mw-redirect" title="Computation of radiowave attenuation in the atmosphere">Computation of radiowave attenuation in the atmosphere</a></li>

<li><a href="Black_hole" title="Black hole">Black hole</a></li></ul>

</td>

<p><b>Numerical algorithms:</b>

</p>

<ul><li><a href="Shooting_method" title="Shooting method">Shooting method</a></li>

<li><a href="Direct_multiple_shooting_method" title="Direct multiple shooting method">Direct multiple shooting method</a></li>

<li><a href="Walk-on-spheres_method" title="Walk-on-spheres method">Walk-on-spheres method</a></li>

<li><a href="Finite_difference_method" title="Finite difference method">Finite difference method</a></li>

<li><a href="Boundary_element_method" title="Boundary element method">Boundary element method</a></li></ul>

<p>

</p>

</td></tr></tbody></table></div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFDaniel_Zwillinger2014" class="citation book cs1">Daniel Zwillinger (12 May 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9QLjBQAAQBAJ&q=%22boundary+value%22&pg=PA536"><i>Handbook of Differential Equations</i></a>. Elsevier Science. pp. 536–. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4832-2096-3</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation book cs1"><i>ISO/IEC/IEEE International Standard - Systems and software engineering</i>. ISO/IEC/IEEE 24765:2010(E). pp. vol., no., pp.1-418.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>A. D. Polyanin and V. F. Zaitsev, <i>Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)</i>, Chapman & Hall/CRC Press, Boca Raton, 2003. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58488-297-2</bdi>.</li>

<li>A. D. Polyanin, <i>Handbook of Linear Partial Differential Equations for Engineers and Scientists</i>, Chapman & Hall/CRC Press, Boca Raton, 2002. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58488-299-9</bdi>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Boundary_value_problems_in_potential_theory">"Boundary value problems in potential theory"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Boundary_value_problem,_complex-variable_methods">"Boundary value problem, complex-variable methods"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><a rel="nofollow" class="external text" href="http://eqworld.ipmnet.ru/en/solutions/lpde.htm">Linear Partial Differential Equations: Exact Solutions and Boundary Value Problems</a> at EqWorld: The World of Mathematical Equations.</li>

<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Boundary_value_problem">"Boundary value problem"</a>. <i><a href="Scholarpedia" title="Scholarpedia">Scholarpedia</a></i>.</cite></li></ul>

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