Study of heat equations with boundary differential equations

In this work we study the numerical solution of one-dimensional heat diffusion equation with a small positive parameter subject to Robin boundary conditions. The simulations examples lead us to conclude that the numerical solutions of the differential equation with Robin boundary condition are very close of the analytic solution of the problem with homogeneous Dirichlet boundary conditions when tends to zero

In this article, a modified cubic B-spline differential quadrature method (MCB-DQM) is proposed to solve some of the basic differential equations. Here we have considered an ordinary differential equation of order two along with heat equation and one- and two-dimensional wave equations. A nonlinear ordinary differential equation of order two is also considered. The ordinary differential equation is reduced to a system of nonhomogeneous linear equations which is then solved by using the Gauss elimination method, whereas the heat equation and the one-dimensional and two-dimensional heat and wave equations are reduced to a system of ordinary differential equations. The system is then solved by the optimal four-stage three-order strong stability preserving time stepping Runge–Kutta (SSP-RK43) scheme. The reliability and efficiency of the method have been tested on six examples.KeywordsOrdinary differential equationHeat equationWave equation cubic B-spline functionsModified cubic B-spline quadrature methodSystem of ordinary differential equationsGauss elimination methodRunge–Kutta fourth-order method

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

Previous chapter Next chapter Other Titles in Applied Mathematics Finite Difference Methods for Ordinary and Partial Differential Equations11. Mixed Equationspp.233 - 242Chapter DOI:https://doi.org/10.1137/1.9780898717839.ch11PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt We have now studied the solution of various types of time-dependent equations: ordinary differential equations (ODEs), parabolic partial differential equations (PDEs) such as the heat equation, and hyperbolic PDEs such as the advection equation. In practice several processes may be happening simultaneously, and the PDE model will not be a pure equation of any of the types already discussed but rather will be a mixture. In this chapter we discuss several approaches to handling more complicated equations. We restrict our attention to time-dependent PDEs of the form ut = A1 (u)+ A2 (u) +⋯+ AN (u) , 11.1 where each of the Aj (u) are (possibly nonlinear) functions or differential operators involving only spatial derivatives of u. For simplicity, most of our discussion will be further restricted to only two terms, which we will write as ut =A (u) +ℬ (u) , 11.2 but more terms often can be handled by extension or combination of the methods described here. 11.1 Some examples We begin with some examples of PDEs involving more than one term. See Appendix E for more discussion of some of these equations. • Multidimensional problems, such as the diffusion equation in two dimensions, ut =κ ( uxx + uyy ) , 11.3 or the three-dimensional version. This problem has already been discussed in Section 9.7, where we saw that efficient methods can be developed by splitting (11.3) into two one-dimensional problems. Previous chapter Next chapter RelatedDetails Published:2007ISBN:978-0-89871-629-0eISBN:978-0-89871-783-9 https://doi.org/10.1137/1.9780898717839Book Series Name:Other Titles in Applied MathematicsBook Code:OT98Book Pages:xv + 328Key words:finite difference methods, stability, accuracy, convergence, ordinary differential equations, partial differential equations, boundary value problems

Previous article Next article On the Transformation of a Class of Boundary Value Problems into Initial Value Problems for Ordinary Differential EquationsMurray S. KlamkinMurray S. Klamkinhttps://doi.org/10.1137/1004006PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. Goldstein, Modern Developments in Fluid Dynamics, Vol. I, Oxford, London, 1957, 135–136 Google Scholar[2] H. P. Greenspan and , G. F. Carrier, The magnetohydrodynamic flow past a flat plate, J. Fluid Mech., 6 (1959), 77–96 MR0108191 0088.19203 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A note on the transformation of boundary value problems to initial value problems: The iterative transformation methodApplied Mathematics and Computation, Vol. 415 | 1 Feb 2022 Cross Ref Sawtooth patterns in flexural force curves of structural biological materials are not signatures of toughness enhancement: Part IIJournal of the Mechanical Behavior of Biomedical Materials, Vol. 124 | 1 Dec 2021 Cross Ref Group invariant solution for a pre-existing fracture driven by a power-law fluid in permeable rockInternational Journal of Modern Physics B, Vol. 30, No. 28n29 | 20 Nov 2016 Cross Ref Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation MethodsActa Applicandae Mathematicae, Vol. 140, No. 1 | 23 October 2014 Cross Ref Propagation of a pre-existing turbulent fluid fractureInternational Journal of Non-Linear Mechanics, Vol. 54 | 1 Sep 2013 Cross Ref An investigation of an Emden-Fowler equation from thin film flowActa Mechanica Sinica, Vol. 28, No. 2 | 28 February 2012 Cross Ref A group invariant solution for a pre-existing fluid-driven fracture in permeable rockNonlinear Analysis: Real World Applications, Vol. 12, No. 1 | 1 Feb 2011 Cross Ref On the equivalence of non-iterative transformation methods based on scaling and spiral groupsMathematical Methods in the Applied Sciences, Vol. 33, No. 5 | 19 June 2009 Cross Ref Analysis of the constant B-number assumption while modeling flame spreadCombustion and Flame, Vol. 152, No. 3 | 1 Feb 2008 Cross Ref Upward flame spread on a vertically oriented fuel surface: The effect of finite widthProceedings of the Combustion Institute, Vol. 31, No. 2 | 1 Jan 2007 Cross Ref Автомодельные решения и степенная геометрияУспехи математических наук, Vol. 55, No. 1 | 1 Jan 2000 Cross Ref BibliographyPower Geometry in Algebraic and Differential Equations | 1 Jan 2000 Cross Ref Self-similar solutionsPower Geometry in Algebraic and Differential Equations | 1 Jan 2000 Cross Ref A Similarity Approach to the Numerical Solution of Free Boundary ProblemsSIAM Review, Vol. 40, No. 3 | 2 August 2006AbstractPDF (387 KB)A Novel Approach to the Numerical Solution of Boundary Value Problems on Infinite IntervalsSIAM Journal on Numerical Analysis, Vol. 33, No. 4 | 12 July 2006AbstractPDF (1468 KB)Nonlinear boundary value problems on semi-infinite intervalsComputers & Mathematics with Applications, Vol. 28, No. 10-12 | 1 Nov 1994 Cross Ref Numerical transformation methods: a constructive approachJournal of Computational and Applied Mathematics, Vol. 50, No. 1-3 | 1 May 1994 Cross Ref The falkneer-skan equation: Numerical solutions within group invariance theoryCalcolo, Vol. 31, No. 1-2 | 1 Mar 1994 Cross Ref Non-iterative Transformation Methods EquivalenceModern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics | 1 Jan 1993 Cross Ref Optimal numerical algorithmsApplied Numerical Mathematics, Vol. 10, No. 3-4 | 1 Sep 1992 Cross Ref Special TopicsNumerical Methods for Partial Differential Equations | 1 Jan 1992 Cross Ref A noniterative transformation method applied to two-point boundary-value problemsApplied Mathematics and Computation, Vol. 39, No. 1 | 1 Sep 1990 Cross Ref A non‐iterative solution of a system of ordinary differential equations arising from boundary layer theoryInternational Journal of Mathematical Education in Science and Technology, Vol. 21, No. 2 | 1 Mar 1990 Cross Ref ReferencesNonlinear Boundary Value Problems in Science and Engineering | 1 Jan 1989 Cross Ref A SOLUTION METHOD FOR A CLASS OF NONLINEAR BOUNDARY VALUE PROBLEMSChemical Engineering Communications, Vol. 43, No. 1-3 | 27 April 2007 Cross Ref Transformation of a Boundary Value Problem to an Initial Value ProblemGroup Invariance in Engineering Boundary Value Problems | 1 Jan 1985 Cross Ref Multiplicity and Stability in Distributed-Parameter Systems (DPS)Computational Methods in Bifurcation Theory and Dissipative Structures | 1 Jan 1983 Cross Ref Supplementary ReferencesGroup Analysis of Differential Equations | 1 Jan 1982 Cross Ref Optimization of nonlinear kinetic equation computationNumerical Integration of Differential Equations and Large Linear Systems | 25 August 2006 Cross Ref Gas discharges in planar low‐pressure thermionic diodes. I. Space‐charge regimeJournal of Applied Physics, Vol. 50, No. 4 | 1 Apr 1979 Cross Ref Non-linear boundary and eigenvalue problems for the emden-fowler equations by group methodsInternational Journal of Non-Linear Mechanics, Vol. 14, No. 1 | 1 Jan 1979 Cross Ref Chapter 1 IntroductionComputational Methods in Engineering Boundary Value Problems | 1 Jan 1979 Cross Ref Chapter 7 Method of Transformation—Direct TransformationComputational Methods in Engineering Boundary Value Problems | 1 Jan 1979 Cross Ref Chapter 8 Method of Transformation—Reduced Physical ParametersComputational Methods in Engineering Boundary Value Problems | 1 Jan 1979 Cross Ref Exact shooting and eigenparameter problemsNonlinear Analysis: Theory, Methods & Applications, Vol. 1, No. 1 | 1 Jan 1976 Cross Ref On the solution of Troesch's nonlinear two-point boundary value problem using an initial value methodJournal of Computational Physics, Vol. 19, No. 3 | 1 Nov 1975 Cross Ref An initial value method for the solution of the magnetohydrodynamic flow past a flat plateActa Physica Academiae Scientiarum Hungaricae, Vol. 36, No. 3 | 1 Apr 1974 Cross Ref A new method for solving eigenvalue problemsJournal of Computational Physics, Vol. 9, No. 1 | 1 Feb 1972 Cross Ref Solving Boundary-Value Problems by ImbeddingJournal of the ACM, Vol. 18, No. 4 | 1 Oct 1971 Cross Ref Transformation of boundary value problems into initial value problemsJournal of Mathematical Analysis and Applications, Vol. 32, No. 2 | 1 Nov 1970 Cross Ref Laminar boundary-layer flows of Newtonian fluids with non-Newtonian fluid injectants (Newtonian fluid laminar boundary layer flow over flat plate with nonNewtonian fluid injection)Journal of Hydronautics, Vol. 4, No. 2 | 1 Apr 1970 Cross Ref An Initial Value Method for the Solution of MHD Boundary-Layer EquationsAeronautical Quarterly, Vol. 21, No. 1 | 7 June 2016 Cross Ref An initial-value method for the solution of certain nonlinear diffusion equations in biologyMathematical Biosciences, Vol. 6 | 1 Jan 1970 Cross Ref An Initial Value Problem Approach to the Solution of Eigenvalue ProblemsSIAM Journal on Numerical Analysis, Vol. 6, No. 1 | 14 July 2006AbstractPDF (434 KB)An efficient higher-order difference method for two-dimensional structures.AIAA Journal, Vol. 7, No. 3 | 1 Mar 1969 Cross Ref High Prandtl number boundary layers with mass injection.AIAA Journal, Vol. 7, No. 3 | 1 Mar 1969 Cross Ref Drag Reduction of a Non-Newtonian Fluid by Fluid Injection at the WallJournal of Hydronautics, Vol. 2, No. 4 | 1 Oct 1968 Cross Ref 3 Examples from Transport PhenomenaNonlinear Ordinary Differential Equations in Transport Processes | 1 Jan 1968 Cross Ref Transforming Boundary Conditions to Initial Conditions for Ordinary Differential EquationsSIAM Review, Vol. 9, No. 2 | 18 July 2006AbstractPDF (486 KB)A Boundary Value ProblemSIAM Review, Vol. 7, No. 4 | 18 July 2006PDF (126 KB)Applications of differential equations in general problem solvingCommunications of the ACM, Vol. 8, No. 9 | 1 Sep 1965 Cross Ref Chapter 6 Further Approximate MethodsNonlinear Partial Differential Equations in Engineering | 1 Jan 1965 Cross Ref Chapter 2: Applications of Modern AlgebraNonlinear Partial Differential Equations in Engineering | 1 Jan 1965 Cross Ref The Finite Deflection of a Normally Loaded, Spinning, Elastic MembraneJournal of the Aerospace Sciences, Vol. 29, No. 10 | 1 Oct 1962 Cross Ref Compressive Stability of Orthotropic CylindersJournal of the Aerospace Sciences, Vol. 29, No. 10 | 1 Oct 1962 Cross Ref Volume 4, Issue 1| 1962SIAM Review1-78 History Submitted:14 August 1961Published online:18 July 2006 InformationCopyright © 1962 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1004006Article page range:pp. 43-47ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

A partial differential equation is an equation which includes derivatives of an unknown function with respect to two or more independent variables. The analytical solution is needed to obtain the exact solution of partial differential equation. To solve these partial differential equations, the appropriate boundary and initial conditions are needed. The general solution is dependent not only on the equation, but also on the boundary conditions. In other words, these partial differential equations will have different general solution when paired with different sets of boundary conditions. In the present study, the homogeneous one-dimensional heat equation will be solved analytically by using separation of variables method. Our main objective is to determine the general and specific solution of heat equation based on analytical solution. To verify our objective, the heat equation will be solved based on the different functions of initial conditions on Neumann boundary conditions. The results have been compared with different values of initial conditions but the boundary condition remain the same. Based on the results obtained, it can be concluded that increase the number of n will reduce the heat temperature and the time taken. For short length of the rod, the heat temperature quickly converges to zero and take less time to release or reduced the heat temperature when compared to the long length of the rod.

Finite difference approximations -- Steady states and boundary value problems -- Elliptic equations -- Iterative methods for sparse linear systems -- The initial value problem for ordinary differential equations -- Zero-stability and convergence for initial value problems -- Absolute stability for ordinary differential equations -- Stiff ordinary differential equations -- Diffusion equations and parabolic problems -- Addiction equations and hyperbolic systems -- Mixed equations -- Appendixes: A. Measuring errors -- B. Polynomial interpolation and orthogonal polynomials -- C. Eigenvalues and inner-product norms -- D. Matrix powers and exponentials -- E. Partial differential equations.

UDC 517.977 We study the problem of optimal control over the processes described by the heat equation and a system of ordinary differential equations. For the problem of optimal control, we prove the existence and uniqueness of solutions, establish sufficient conditions for the Frechet differentiability of the purpose functional, deduce the expression for its gradient, and obtain necessary conditions of optimality in the form of an integral maximum principle. sources is considered for systems described only by the heat equation. In addition, the cited works deal only with the systems with distributed parameters. At the same time, in the construction of mathematical models of various dynamical systems, it is necessary to take into account auxiliary elements without which it is impossible to realize control over the analyzed process. As a rule, these elements have concentrated parameters. The behavior of these systems is described by a collection of ordinary and partial differential equations with initial and boundary conditions. In the present paper, we consider the variational method for the solution of the problem of optimal control over moving sources in the form of the heat equation and a system of ordinary differential equations with initial and boundary conditions. For this problem, we prove the theorem on existence and uniqueness of the solution, establish sufficient conditions for the Frechet differentiability of the objective functional, and deduce the expres-

PurposeThis study aims to present a numerical method for solving the one-dimensional heat equation with temperature-dependent thermal conductivity. General nonlinear boundary conditions that can depend on time explicitly are considered.Design/methodology/approachFirst, using an implicit scheme, the heat equation is discretized in time, whereby, at each time level, a nonlinear two-point boundary value problem (TPBVP) is obtained. To solve the nonlinear TPBVPs, the quasilinearization method is applied. The obtained linear sub-problems are solved by the finite difference method.FindingsThe whole method is unconditionally stable. Its computational efficiency is high. The time complexity of the algorithm is O(MN), where M is the number of time levels and N is the number of space mesh points. Examples with exponential and power law dependence of the thermal diffusivity on temperature and different boundary conditions, including fixed temperature, fixed flux, convection, relaxing and oscillating conditions, are presented. The results confirm the unconditional stability of the method and its high computational efficiency.Practical implications In addition to being unconditionally stable and computationally very efficient, the proposed method is quite easy to implement. This is demonstrated by the provided four MATLAB codes. They treat different types of boundary conditions and different dependences of the thermal conductivity on the temperature.Originality/valueThe novelty of the method lies in converting the heat equation, which is a partial differential equation, into a sequence of ordinary differential equations (ODEs) with boundary conditions. This allows using methods for ODEs.

Radiative frost is one of the most severe weather conditions that affects agricultural activities in many parts of the world. Since various protective methods to reduce frost impact are available, refinements of frost forecasting methodologies should provide economical benefits. In the present study, a three-dimensional numerical local-scale model for the simulation of the microclimate near the ground surface of nonhomogeneous regions during radiative frost events was developed. The model is based on the equations of motion, heat, humidity and continuity in the atmosphere and the equations of heat and moisture diffusion in the soil. Emphasis was given in establishing a refined formulation of energy budget equations for soil surface and plant canopy Additionally, an improved finite difference scheme procedure for approximating horizontal derivatives in a terrain-following coordinate system was introduced. The sensitivity of the model to various parameters that way affect the nocturnal minimum temperature near ground surface during radiative frost events was tested by using one- and two-dimensional versions of the model. This temperature was found to be sensitive to topography, plant cover, soil moisture content, air specific humidity and wind velocity.

<abstract><p>Within the framework of time fractional calculus using the Caputo operator, the Aboodh residual power series method and the Aboodh transform iterative method were implemented to analyze three basic equations in mathematical physics: the heat equation, the diffusion equation, and Burger's equation. We investigated the analytical solutions of these equations using Aboodh techniques, which provide practical and precise methods for solving fractional differential equations. We clarified the behavior and properties of the obtained approximations using the suggested methods through exact mathematical derivations and computational analysis. The obtained approximations were analyzed numerically and graphically to verify their high accuracy and stability against different related parameters. Additionally, we examined the impact of varying the fractional parameter the profiles of all derived approximations. Our results confirm these methods, efficacy in capturing the complicated dynamics of fractional systems. Therefore, they enhance the comprehension and examination of time-fractional equations in many scientific and technical contexts and in modeling different physical problems related to fluid mediums and plasma physics.</p></abstract>

Previous article Next article Determination of an Unknown Heat Source from Overspecified Boundary DataJ. R. CannonJ. R. Cannonhttps://doi.org/10.1137/0705024PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. R. Cannon, Determination of an unknown coefficient in a parabolic differential equation, Duke Math. J., 30 (1963), 313–323 10.1215/S0012-7094-63-03033-3 MR0157121 (28:358) 0117.06901 CrossrefISIGoogle Scholar[2] J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188–201 10.1016/0022-247X(64)90061-7 MR0160047 (28:3261) 0131.32104 CrossrefGoogle Scholar[3] J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla \cdot k(u)\nabla u=0$ from overspecified boundary data, J. Math. Anal. Appl., 18 (1967), 112–114 10.1016/0022-247X(67)90185-0 MR0209634 (35:531) 0151.15901 CrossrefISIGoogle Scholar[4] J. R. Cannon and , D. L. Filmer, The determination of unknown parameters in analytic systems of ordinary differential equations, SIAM J. Appl. Math., 15 (1967), 799–809 10.1137/0115069 MR0218632 (36:1716) 0251.34002 LinkISIGoogle Scholar[5] J. R. Cannon, , Jim Douglas, Jr. and , B. Frank Jones, Jr., Determination of the diffusivity of an isotropic medium, Internat. J. Engrg. Sci., 1 (1963), 453–455 10.1016/0020-7225(63)90002-8 MR0160045 (28:3259) CrossrefGoogle Scholar[6] J. R. Cannon and , B. Frank Jones, Jr., Determination of the diffusivity of an anisotropic medium, Internat. J. Engrg. Sci., 1 (1963), 457–460 10.1016/0020-7225(63)90003-X MR0160046 (28:3260) CrossrefGoogle Scholar[7] J. R. Cannon and , J. H. Halton, The irrotational solution of an elliptic differential equation with an unknown coefficient, Proc. Cambridge Philos. Soc., 59 (1963), 680–682 MR0149064 (26:6560) 0117.07101 CrossrefISIGoogle Scholar[8] Jim Douglas, Jr. and , B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. II. Numerical approximation, J. Math. Mech., 11 (1962), 919–926 MR0153988 (27:3949) 0112.32603 ISIGoogle Scholar[9] B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), 907–918 MR0153987 (27:3948) 0112.32602 ISIGoogle Scholar[10] B. Frank Jones, Jr., Various methods for finding unknown coefficients in parabolic differential equations, Comm. Pure Appl. 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MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705024Article page range:pp. 275-286ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

This chapter applies the theory of linear ordinary differential equations to certain boundary-value problems for partial differential equations. Section 1 briefly introduces some notation and defines the three partial differential equations of principal interest—the heat equation, Laplace’s equation, and the wave equation. Section 2 is a first exposure to solving partial differential equations, working with boundary-value problems for the three equations introduced in Section 1. The settings are ones where the method of “separation of variables” is successful. In each case the equation reduces to an ordinary differential equation in each independent variable, and some analysis is needed to see when the method actually solves a particular boundary-value problem. In simple cases Fourier series can be used. In more complicated cases Sturm’s Theorem, which is stated but not proved in this section, can be helpful. Section 3 returns to Sturm’s Theorem, giving a proof contingent on the Hilbert–Schmidt Theorem, which itself is proved in Chapter II. The construction within this section finds a Green’s function for the second-order ordinary differential operator under study; the Green’s function defines an integral operator that is essentially an inverse to the second-order differential operator.

This chapter applies the theory of linear ordinary differential equations to certain boundary-value problems for partial differential equations.Section 1 briefly introduces some notation and defines the three partial differential equations of principal interest—the heat equation, Laplace’s equation, and the wave equation.Section 2 is a first exposure to solving partial differential equations, working with boundary-value problems for the three equations introduced in Section 1. The settings are ones where the method of “separation of variables” is successful. In each case the equation reduces to an ordinary differential equation in each independent variable, and some analysis is needed to see when the method actually solves a particular boundary-value problem. In simple cases Fourier series can be used. In more complicated cases Sturm’s Theorem, which is stated but not proved in this section, can be helpful.Section 3 returns to Sturm’s Theorem, giving a proof contingent on the Hilbert-Schmidt Theorem, which itself is proved in Chapter II. The construction within this section finds a Green’s function for the second-order ordinary differential operator under study; the Green’s function defines an integral operator that is essentially an inverse to the second-order differential operator.KeywordsHeat EquationBoundary DataContinuous DerivativeNonzero SolutionLiouville TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.