Variations of heat equation on the half‐line via the Fokas method

A novel technique is presented for explicitly solving inhomogeneous initial‐boundary‐value problems (IBVPs) (Dirichlet, Neumann and Robin) on the half‐line, for a well‐known pseudo‐parabolic partial differential equation. This so‐called Barenblatt's equation arises in a plethora of important applications, ranging from heat‐mass transfer, solid‐fluid‐gas dynamics and materials science, to mechanical, chemical and petroleum engineering, as well as electron physics, radiation and diffusive processes. Our approach is based on the extension of the Fokas method, so that it can be applied to problems with mixed derivatives. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each type of IBVP, the novel formulae are utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain and the problem's well‐posedness. Furthermore, importantly, our solutions’ numerical advantages are demonstrated and highlighted by way of a concrete and illustrative example. Our rigorous approach can be extended to IBVPs for other significant models.

Separation of variables... Fourier transforms... Green's functions... The Eastern cut-off... There are many techniques in the undergraduate canon for solving linear partial differential equations (PDEs) that have been popularized in excellent introductory texts. I confess that “The Eastern Cut-off” is not a mathematical technique, but rather, it was a well-regarded method for the high jump. By coincidence, the early 1800s saw both the first publication of Joseph Fourier's investigations into solutions to the heat equation using trigonometric series and the introduction of the high jump as an athletic event. For more than 150 years, the Eastern cut-off and a collection of other approaches were “known” to be the best high jump techniques until Richard Fosbury shocked the track and field community by turning away from the bar and leaping over it backwards with better results than other athletes could achieve. The lesson is clear: Success can discourage innovation by leading one to believe that the current workable solution is the best and only solution. This section of SIAM Review addresses this issue by injecting new ideas into traditional courses, and “The Method of Fokas for Solving Linear Partial Differential Equations,” by Bernard Deconinck, Thomas Trogdon, and Vishal Vasan, is ideally suited to this purpose. Fokas's method is a relatively new approach that should supplement the existing basket of methods and techniques, including separation of variables, Fourier transform, Laplace transforms, and so forth. In the context of this tutorial, a solution is an explicit expression that solves the PDE and satisfies the given boundary and initial conditions. The explicit expression typically takes the form of integrals or summations involving these boundary and initial conditions. (An explicit expression composed of information that is not provided in the statement of the problem, such as Dirichlet in place of Neumann boundary data, would not be considered a solution.) The new approach is to transform the linear PDE into a local relation of the form \[ \frac\partial \rho\partial t + \frac\partial j\partial x = 0, \rho(x,t,k) = e^-ikx + ømega(k)t q(x,t), \] where $q(x,t)$ is the dependent variable in the PDE, $\omega(k)$ is the dispersion relation, and $j(x,t,k)$ is deduced from the PDE. To solve the PDE, one applies Green's Theorem to the local relation over the $(x,t)$ domain with boundaries that fit the initial and boundary data. The majority of the effort involves mapping the resulting line integrals into the available information along the boundaries of the domain. In a broader sense, more traditional integral transform techniques are suited to certain types of boundary data, while the method of Fokas is a route to an appropriate integral transform by mathematical inspection and manipulation. As such, it can be a more general approach to a broad category of problems. There are advantages and disadvantages to covering this approach in the undergraduate curriculum. The method of Fokas requires students to have a solid background in complex analysis in order to know which manipulations and symmetries will map the existing integral expressions to the initial and boundary information posed in the problem. The traditional methods in a first course in PDEs require little background beyond a course in ODEs, but at the same time these methods are limited to a small set of problems. For an undergraduate PDEs course in which students have had some complex analysis beforehand, or for a graduate course, the method of Fokas is an outstanding supplement that provides a fresh perspective on integral transforms and where one transform might be more useful than another. This particular tutorial is limited to constant coefficient evolution equations with one spatial dimension, but there are many extensions and potential student projects available by digging into the literature. Similar to the “Fosbury Flop,” the method of Fokas approaches familiar problems from a new direction, providing students and instructors with new insights into linear PDEs.

Solving the heat equation with variable thermal conductivity

We examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation for solutions to half-line and finite-interval initial-boundary value problems as integrals of kernels with explicit spatial and temporal dependence. These solution representations are defined within the spatial domain of the problem. We obtain the extension of these representation formulae via Taylor series outside these spatial domains and find the extension of the initial condition that gives rise to a whole-line initial-value problem solved by the extended solution. In general, the extended initial condition is not differentiable or continuous unless the boundary and initial conditions satisfy compatibility conditions. We analyse dissipative and dispersive problems, and problems with continuous and discrete spatial variables.

We propose a new method to solve the initial-boundary value problem for hyperbolic-dissipative partial differential equations (PDEs) based on the spirit of LY algorithm [T.-P. Liu and S.-H. Yu, Dirichlet–Neumann kernel for hyperbolic-dissipative system in half-space, Bull. Inst. Math. Acad. Sin. 7 (2012) 477–543]. The new method can handle more general domains than that of LYs’. We convert the evolutionary PDEs into the elliptic PDEs by the Laplace transformation. Using the Laplace transformation of the fundamental solutions of the evolutionary PDEs and the image method, we can construct Green’s functions for the corresponding elliptic PDEs. Finally, we obtain Green’s functions for the evolutionary PDEs by inverting the Laplace transformation. As a consequence, we establish Green’s functions for some basic PDEs such as the heat equation, the wave equation and the damped wave equation, in a half space and a quarter plane with various boundary conditions. On the other hand, the structure of hyperbolic-dissipative PDEs means its fundamental solution is non-symestric and hence the image method does not work. We utilize the idea of Laplace wave train introduced by Liu and Yu in [Navier–Stokes equations in gas dynamics: Greens function, singularity and well-posedness, Comm. Pure Appl. Math. 75(2) (2022) 223–348] to generalize the image method. Combining this with the notions of Rayleigh surface wave operators introduced in [S. J. Deng, W. K. Wang and S.-H. Yu, Green’s functions of wave equations in [Formula: see text], Arch. Ration. Mech. Anal. 216 (2015) 881–903], we are able to obtain the complete representations of Green’s functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions.

In this short communication, we announce an algorithmic procedure for constructing non‐uniqueness counter‐examples of classical solutions to initial‐boundary‐value problems for a wide class of linear evolution partial differential equations (PDE), of any order and with constant coefficients, formulated in a quarter‐plane. Our approach relies on analysis of regularity and asymptotic properties, near the boundary of the spatio‐temporal domain, of closed‐form integral‐representation formulae derived via complex‐analytic techniques and rigorous implementation of the modern PDE technique known as Fokas unified transform method. In order to elucidate the novel idea and demonstrate the proposed technique in a self‐contained fashion, we explicitly present its application to two concrete examples, namely the heat equation and the linear KdV equation with Dirichlet data. New uniqueness theorems for these two models are also presented herein.

Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.

The unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.

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. Math., 16 (1963), 33–44 MR0152760 (27:2735) 0119.08302 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Identifying a space-dependent source term in distributed order time-fractional diffusion equationsMathematical Control and Related Fields, Vol. 0, No. 0 | 1 Jan 2022 Cross Ref Identification of stationary source in the anomalous diffusion equationInverse Problems in Science and Engineering, Vol. 29, No. 13 | 21 November 2021 Cross Ref A modified quasi-reversibility method for inverse source problem of Poisson equationInverse Problems in Science and Engineering, Vol. 29, No. 12 | 22 March 2021 Cross Ref Inverse modeling of contaminant transport for pollution source identification in surface and groundwaters: a reviewGroundwater for Sustainable Development, Vol. 15 | 1 Nov 2021 Cross Ref Convergence Analysis of a Crank–Nicolson Galerkin Method for an Inverse Source Problem for Parabolic Equations with Boundary ObservationsApplied Mathematics & Optimization, Vol. 84, No. 2 | 10 August 2020 Cross Ref Space-time finite element method for determination of a source in parabolic equations from boundary observationsJournal of Inverse and Ill-posed Problems, Vol. 29, No. 5 | 6 August 2020 Cross Ref Identification of the unknown heat source terms in a 2D parabolic equationJournal of King Saud University - Science, Vol. 33, No. 6 | 1 Sep 2021 Cross Ref Inverse source problem for a space-time fractional diffusion equationRicerche di Matematica, Vol. 74 | 14 August 2021 Cross Ref Identifying an unknown source term in a time-space fractional parabolic equationApplied Numerical Mathematics, Vol. 166 | 1 Aug 2021 Cross Ref Regularization method for the problem of determining the source function using integral conditionsAdvances in the Theory of Nonlinear Analysis and its Application | 6 May 2021 Cross Ref Identifying an unknown source term in a heat equation with time-dependent coefficientsInverse Problems in Science and Engineering, Vol. 29, No. 5 | 29 July 2020 Cross Ref Data completion problem for the advection‐diffusion equation with aquifer point sourcesMathematical Methods in the Applied Sciences, Vol. 44, No. 2 | 27 September 2020 Cross Ref Inverse Heat Source Problem and Experimental Design for Determining Iron Loss DistributionAntti Hannukainen, Nuutti Hyvönen, and Lauri PerkkiöSIAM Journal on Scientific Computing, Vol. 43, No. 2 | 1 March 2021AbstractPDF (4389 KB)Identification of a Time Dependent Source Function in a Parabolic Inverse Problem Via Finite Element ApproachIndian Journal of Pure and Applied Mathematics, Vol. 51, No. 4 | 5 January 2021 Cross Ref Detection of time-varying heat sources using an analytic forward modelJournal of Computational and Applied Mathematics, Vol. 379 | 1 Dec 2020 Cross Ref Solving a source distribution in heat conduction equation by homotopy analysis methodJournal of Physics: Conference Series, Vol. 1707, No. 1 | 1 Nov 2020 Cross Ref Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguideApplicable Analysis, Vol. 99, No. 13 | 18 December 2018 Cross Ref Solution of the Boundary-Value Problem of Heat Conduction with Periodic Boundary ConditionsUkrainian Mathematical Journal, Vol. 72, No. 2 | 4 September 2020 Cross Ref A hybrid scheme for time fractional inverse parabolic problemWaves in Random and Complex Media, Vol. 30, No. 2 | 27 August 2018 Cross Ref Inverse source problem for a distributed-order time fractional diffusion equationJournal of Inverse and Ill-posed Problems, Vol. 28, No. 1 | 13 June 2019 Cross Ref On the Identification of Source Term in the Heat Equation from Sparse DataWilliam Rundell and Zhidong ZhangSIAM Journal on Mathematical Analysis, Vol. 52, No. 2 | 30 March 2020AbstractPDF (1534 KB)Determination of an Unknown Time-dependent Heat Source from A Nonlocal Measurement by Finite Difference MethodActa Mathematicae Applicatae Sinica, English Series, Vol. 36, No. 1 | 27 December 2019 Cross Ref A Tikhonov Regularization Method for Solving an Inverse Heat Source ProblemBulletin of the Malaysian Mathematical Sciences Society, Vol. 43, No. 1 | 23 October 2018 Cross Ref Two-dimensional inverse quasilinear parabolic problem with periodic boundary conditionApplicable Analysis, Vol. 98, No. 8 | 6 February 2018 Cross Ref Simultaneous determination of the heat source and the initial data by using an explicit Lie-group shooting methodNumerical Heat Transfer, Part B: Fundamentals, Vol. 75, No. 4 | 13 June 2019 Cross Ref Reversing Inverse Problem of Source Term of Heat Conduction EquationAdvances in Applied Mathematics, Vol. 08, No. 01 | 1 Jan 2019 Cross Ref The method of fundamental solution for the inverse source problem for the space-fractional diffusion equationInverse Problems in Science and Engineering, Vol. 26, No. 7 | 4 September 2017 Cross Ref Parameter identification by optimization method for a pollution problem in porous mediaActa Mathematica Scientia, Vol. 38, No. 4 | 1 Jul 2018 Cross Ref Identification of a multi-dimensional space-dependent heat source from boundary dataApplied Mathematical Modelling, Vol. 54 | 1 Feb 2018 Cross Ref A meshless method for solving 1D time-dependent heat source problemInverse Problems in Science and Engineering, Vol. 26, No. 1 | 26 April 2017 Cross Ref Conjugate Gradient Method for Identification of a Spacewise Heat SourceLarge-Scale Scientific Computing | 3 January 2018 Cross Ref Numerical reconstruction of the right hand side with separable variables of the parabolic equation1 Jan 2018 Cross Ref Source degenerate identification problems with smoothing overdeterminationAdvances in Difference Equations, Vol. 2017, No. 1 | 26 October 2017 Cross Ref Detection-Identification of multiple unknown time-dependent point sources in a 2 D transport equation: application to accidental pollutionInverse Problems in Science and Engineering, Vol. 25, No. 10 | 20 December 2016 Cross Ref A Numerical Algorithm Based on RBFs for Solving an Inverse Source ProblemBulletin of the Malaysian Mathematical Sciences Society, Vol. 40, No. 3 | 30 March 2016 Cross Ref Detection and identification of multiple unknown time-dependent point sources occurring in 1D evolution transport equationsInverse Problems in Science and Engineering, Vol. 25, No. 4 | 8 April 2016 Cross Ref A meshless method to the numerical solution of an inverse reaction–diffusion–convection problemInternational Journal of Computer Mathematics, Vol. 94, No. 3 | 10 May 2016 Cross Ref Identification of time-dependent source terms and control parameters in parabolic equations from overspecified boundary dataJournal of Computational and Applied Mathematics, Vol. 313 | 1 Mar 2017 Cross Ref Determination of a term in the right-hand side of parabolic equationsJournal of Computational and Applied Mathematics, Vol. 309 | 1 Jan 2017 Cross Ref On the Inverse Problem of Dupire’s Equation with Nonlocal Boundary and Integral ConditionsJournal of Mathematical Finance, Vol. 07, No. 04 | 1 Jan 2017 Cross Ref Inverse Problem for a Weakly Nonlinear Ultraparabolic Equation with Three Unknown Functions of Different Arguments on the Right-Hand SideJournal of Mathematical Sciences, Vol. 217, No. 4 | 23 July 2016 Cross Ref An inverse source problem in a semilinear time-fractional diffusion equationComputers & Mathematics with Applications, Vol. 72, No. 6 | 1 Sep 2016 Cross Ref Determination of a Time-Dependent Term in the Right-Hand Side of Linear Parabolic EquationsActa Mathematica Vietnamica, Vol. 41, No. 2 | 9 July 2015 Cross Ref Identification of time active limit with lower and upper bounds of total amount loaded by unknown sources in 2D transport equationsJournal of Engineering Mathematics, Vol. 97, No. 1 | 28 July 2015 Cross Ref A new approximate method for an inverse time-dependent heat source problem using fundamental solutions and RBFsEngineering Analysis with Boundary Elements, Vol. 64 | 1 Mar 2016 Cross Ref Source Localization of Reaction-Diffusion Models for Brain TumorsPattern Recognition | 27 August 2016 Cross Ref Recovery of a time dependent source from a surface measurement in Maxwell’s equationsComputers & Mathematics with Applications, Vol. 71, No. 1 | 1 Jan 2016 Cross Ref Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equationsJournal of Inverse and Ill-posed Problems, Vol. 24, No. 6 | 1 Jan 2016 Cross Ref Numerical simulation for an inverse source problem in a degenerate parabolic equationApplied Mathematical Modelling, Vol. 39, No. 23-24 | 1 Dec 2015 Cross Ref Determination of a solely time-dependent source in a semilinear parabolic problem by means of boundary measurementsJournal of Computational and Applied Mathematics, Vol. 289 | 1 Dec 2015 Cross Ref Fourier collocation algorithm for identifying the spacewise-dependent source in the advection–diffusion equation from boundary data measurementsApplied Numerical Mathematics, Vol. 97 | 1 Nov 2015 Cross Ref A parabolic inverse source problem with a dynamical boundary conditionApplied Mathematics and Computation, Vol. 256 | 1 Apr 2015 Cross Ref An iterative algorithm for identifying heat source by using a DQ and a Lie-group methodInverse Problems in Science and Engineering, Vol. 23, No. 1 | 30 January 2014 Cross Ref Reconstruction of a Robin Coefficient by a Predictor-Corrector MethodMathematical Problems in Engineering, Vol. 2015 | 1 Jan 2015 Cross Ref Identifying a time-dependent heat source with nonlocal boundary and overdetermination conditions by the variational iteration methodInternational Journal of Numerical Methods for Heat & Fluid Flow, Vol. 24, No. 7 | 26 Aug 2014 Cross Ref Recovering the source and initial value simultaneously in a parabolic equationInverse Problems, Vol. 30, No. 6 | 2 June 2014 Cross Ref Identification and reconstruction of elastic body forcesInverse Problems, Vol. 30, No. 5 | 1 May 2014 Cross Ref Some uniqueness theorems for inverse spacewise dependent source problems in nonlinear PDEsInverse Problems in Science and Engineering, Vol. 22, No. 1 | 5 August 2013 Cross Ref Moving Least Squares Method for a One-Dimensional Parabolic Inverse ProblemAbstract and Applied Analysis, Vol. 2014 | 1 Jan 2014 Cross Ref Inverse source problem in a one-dimensional evolution linear transport equation with spatially varying coefficients: application to surface water pollutionInverse Problems in Science and Engineering, Vol. 21, No. 6 | 18 February 2013 Cross Ref Numerical identification of source terms for a two dimensional heat conduction problem in polar coordinate systemApplied Mathematical Modelling, Vol. 37, No. 3 | 1 Feb 2013 Cross Ref Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient methodComputers & Mathematics with Applications, Vol. 65, No. 1 | 1 Jan 2013 Cross Ref Identification of the pollution source from one-dimensional parabolic equation modelsApplied Mathematics and Computation, Vol. 219, No. 8 | 1 Dec 2012 Cross Ref Reconstruction of a space and time dependent heat source from finite measurement dataInternational Journal of Heat and Mass Transfer, Vol. 55, No. 23-24 | 1 Nov 2012 Cross Ref Identification of an unknown source depending on both time and space variables by a variational methodApplied Mathematical Modelling, Vol. 36, No. 10 | 1 Oct 2012 Cross Ref Identification of multiple moving pollution sources in surface waters or atmospheric media with boundary observationsInverse Problems, Vol. 28, No. 7 | 25 June 2012 Cross Ref Identifying an unknown source term in radial heat conductionInverse Problems in Science and Engineering, Vol. 20, No. 3 | 12 October 2011 Cross Ref Inverse source problem in a 2D linear evolution transport equation: detection of pollution sourceInverse Problems in Science and Engineering, Vol. 20, No. 3 | 22 November 2011 Cross Ref A Lie-group shooting method for reconstructing a past time-dependent heat sourceInternational Journal of Heat and Mass Transfer, Vol. 55, No. 5-6 | 1 Feb 2012 Cross Ref Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditionsApplied Mathematics and Computation, Vol. 218, No. 8 | 1 Dec 2011 Cross Ref The determination of heat sources in two dimensional inverse steady heat problems by means of the method of fundamental solutionsInverse Problems in Science and Engineering, Vol. 19, No. 6 | 11 March 2011 Cross Ref Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimizationInternational Journal of Heat and Mass Transfer, Vol. 54, No. 17-18 | 1 Aug 2011 Cross Ref Inverse problem of time-dependent heat sources numerical reconstructionMathematics and Computers in Simulation, Vol. 81, No. 8 | 1 Apr 2011 Cross Ref A direct numerical method for solving inverse heat source problemsJournal of Physics: Conference Series, Vol. 290 | 5 May 2011 Cross Ref A self-adaptive LGSM to recover initial condition or heat source of one-dimensional heat conduction equation by using only minimal boundary thermal dataInternational Journal of Heat and Mass Transfer, Vol. 54, No. 7-8 | 1 Mar 2011 Cross Ref Appendix B: References to the Literature on the IHCPThe Mollification Method and the Numerical Solution of Ill-Posed Problems | 22 February 2011 Cross Ref Identification of moving pointwise sources in an advection–dispersion–reaction equationInverse Problems, Vol. 27, No. 2 | 18 January 2011 Cross Ref Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problemComputer Physics Communications, Vol. 181, No. 12 | 1 Dec 2010 Cross Ref Hölder-Type Approximation for the Spatial Source Term of a Backward Heat EquationNumerical Functional Analysis and Optimization, Vol. 31, No. 12 | 19 Nov 2010 Cross Ref Source term identification for an axisymmetric inverse heat conduction problemComputers & Mathematics with Applications, Vol. 59, No. 1 | 1 Jan 2010 Cross Ref A method for identifying a spacewise-dependent heat source under stochastic noise interferenceInverse Problems in Science and Engineering, Vol. 18, No. 1 | 4 November 2009 Cross Ref Identification of a time-varying point source in a system of two coupled linear diffusion-advection- reaction equations: application to surface water pollutionInverse Problems, Vol. 25, No. 11 | 29 October 2009 Cross Ref The method of fundamental solutions for the inverse space-dependent heat source problemEngineering Analysis with Boundary Elements, Vol. 33, No. 10 | 1 Oct 2009 Cross Ref Optimization method for the inverse problem of reconstructing the source term in a parabolic equationMathematics and Computers in Simulation, Vol. 80, No. 2 | 1 Oct 2009 Cross Ref Two regularization strategies for an evolutional type inverse heat source problemJournal of Physics A: Mathematical and Theoretical, Vol. 42, No. 36 | 19 August 2009 Cross Ref The recovery of a time-dependent point source in a linear transport equation: application to surface water pollutionInverse Problems, Vol. 25, No. 7 | 1 June 2009 Cross Ref Determine the special term of a two-dimensional heat sourceApplicable Analysis, Vol. 88, No. 3 | 1 Mar 2009 Cross Ref The method of fundamental solutions for the inverse heat source problemEngineering Analysis with Boundary Elements, Vol. 32, No. 3 | 1 Mar 2008 Cross Ref A procedure for determining a spacewise dependent heat source and the initial temperatureApplicable Analysis, Vol. 87, No. 3 | 1 Mar 2008 Cross Ref Determination of a spacewise dependent heat sourceJournal of Computational and Applied Mathematics, Vol. 209, No. 1 | 1 Dec 2007 Cross Ref Identification of point sources in two-dimensional advection-diffusion-reaction equation: application to pollution sources in a river. Stationary caseInverse Problems in Science and Engineering, Vol. 15, No. 8 | 18 December 2007 Cross Ref The boundary-element method for the determination of a heat source dependent on one variableJournal of Engineering Mathematics, Vol. 54, No. 4 | 3 January 2006 Cross Ref Inverse source problem for a transmission problem for a parabolic equationJournal of Inverse and Ill-posed Problems, Vol. 14, No. 1 | 1 Jan 2006 Cross Ref Identification of a point source in a linear advection–dispersion–reaction equation: application to a pollution source problemInverse Problems, Vol. 21, No. 3 | 9 May 2005 Cross Ref Conditional stability in determining a heat sourceJournal of Inverse and Ill-posed Problems, Vol. 12, No. 3 | 1 Jun 2004 Cross Ref Lipschitz stability in inverse parabolic problems by the Carleman estimateInverse Problems, Vol. 14, No. 5 | 1 January 1999 Cross Ref Determination of the unknown sources in the heat-conduction equationComputational Mathematics and Modeling, Vol. 8, No. 4 | 1 Oct 1997 Cross Ref An estimation method for point sources of multidimensional diffusion equationApplied Mathematical Modelling, Vol. 21, No. 2 | 1 Feb 1997 Cross Ref A reliable estimation method for locations of point sources for an n-dimensional Poisson equationApplied Mathematical Modelling, Vol. 20, No. 11 | 1 Nov 1996 Cross Ref Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary dataInverse Problems, Vol. 10, No. 6 | 1 January 1999 Cross Ref Conditional Stability in Determination of Densities of Heat Sources in a Bounded DomainControl and Estimation of Distributed Parameter Systems: Nonlinear Phenomena | 1 Jan 1994 Cross Ref Conditional stability in determination of force terms of heat equations in a rectangleMathematical and Computer Modelling, Vol. 18, No. 1 | 1 Jul 1993 Cross Ref Inverse problems and Carleman estimatesInverse Problems, Vol. 8, No. 4 | 1 January 1999 Cross Ref Determination of an unknown function in a parabolic equation with an overspecified conditionMathematical Methods in the Applied Sciences, Vol. 13, No. 5 | 1 Nov 1990 Cross Ref Some stability estimates for a heat source in terms of overspecified data in the 3-D heat equationJournal of Mathematical Analysis and Applications, Vol. 147, No. 2 | 1 Apr 1990 Cross Ref Parameter identification in hyperbolic and parabolic partial differential equations of cylindrical geometry from overspecified boundary dataInternational Journal of Engineering Science, Vol. 28, No. 10 | 1 Jan 1990 Cross Ref Identification of certain physical parameters in hyperbolic boundary value problems from overspecified boundary dataInternational Journal of Engineering Science, Vol. 26, No. 8 | 1 Jan 1988 Cross Ref An Inverse Problem for a Nonlinear Elliptic Differential EquationMichael Pilant and William RundellSIAM Journal on Mathematical Analysis, Vol. 18, No. 6 | 1 August 2006AbstractPDF (793 KB)An inverse problem for the heat equationInverse Problems, Vol. 2, No. 4 | 1 January 1999 Cross Ref A Note on an Inverse Problem Related to the 3-D Heat EquationInverse Problems | 1 Jan 1986 Cross Ref Determination of a Source Term in a Linear Parabolic Differential Equation with Mixed Boundary ConditionsInverse Problems | 1 Jan 1986 Cross Ref Nonparametric algorithm for input signals identification in static distributed-parameter systemsIEEE Transactions on Automatic Control, Vol. 29, No. 7 | 1 Jul 1984 Cross Ref Parameter determination in parabolic partial differential equations from overspecified boundary dataInternational Journal of Engineering Science, Vol. 20, No. 6 | 1 Jan 1982 Cross Ref Determination of an unknown non-homogeneous term in a linear partial differential equation .from overspecified boundary dataApplicable Analysis, Vol. 10, No. 3 | 2 May 2007 Cross Ref Distributed parameter system indentification A survey†International Journal of Control, Vol. 26, No. 4 | 16 May 2007 Cross Ref Determination of a source term in a linear parabolic partial differential equationZeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 27, No. 3 | 1 May 1976 Cross Ref Some general remarks on improperly posed problems for partial differential equationsSymposium on Non-Well-Posed Problems and Logarithmic Convexity | 22 August 2006 Cross Ref Estimation of parameters in partial differential equations from noisy experimental dataChemical Engineering Science, Vol. 26, No. 6 | 1 Jun 1971 Cross Ref Determination of an unknown forcing function in a hyperbolic equation from overspecified dataAnnali di Matematica Pura ed Applicata, Vol. 85, No. 1 | 1 Dec 1970 Cross Ref The character of non-uniqueness in the conductivity modelling problem for the earthPure and Applied Geophysics PAGEOPH, Vol. 80, No. 1 | 1 Jan 1970 Cross Ref Volume 5, Issue 2| 1968SIAM Journal on Numerical Analysis199-459 History Submitted:29 September 1967Published online:03 August 2006 InformationCopyright © 1968 © Society for Industrial and Applied 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

A higher dispersion KdV equation on the half-line

In this paper, we solve explicitly and analyze rigorously inhomogeneous initial-boundary-value problems (IBVP) for several fourth-order variations of the traditional diffusion equation and the associated linearized Cahn–Hilliard (C-H) model (also Kuramoto–Sivashinsky equation), formulated in the spatiotemporal quarter-plane. Such models are of relevance to heat-mass transfer phenomena, solid-fluid dynamics and the applied sciences. In particular, we derive formally effective solution representations, justifying a posteriori their validity. This includes the reconstruction of the prescribed initial and boundary data, which requires 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 is utilized to rigorously deduce the solution’s regularity and asymptotic properties near the boundaries of the domain, including uniform convergence, eventual (long-time) periodicity under (eventually) periodic boundary conditions, and null noncontrollability. Importantly, this analysis is indispensable for exploring the (non)uniqueness of the problem’s solution and a new counter-example is constructed. Our work is based on the synergy between: (i) the well-known Fokas unified transform method and (ii) a new approach recently introduced for the rigorous analysis of the Fokas method and for investigating qualitative properties of linear evolution partial differential equations (PDE) on semi-infinite strips. Since only up to third-order evolution PDE have been investigated within this novel framework to date, we present our analysis and results in an illustrative manner and in order of progressively greater complexity, for the convenience of readers. The solution formulae established herein are expected to find utility in well-posedness and asymptotics studies for nonlinear counterparts too.

This work investigates the initial‐boundary value problem (IBVP) of the Klein–Gordon (KG) equation on the half‐line within the Sobolev spaces framework. By employing the Fokas method coupled with the Banach fixed‐point theorem, we establish the following key results: (i) For the IBVP of linear KG equation, we prove the well‐posedness results through decomposition into a free Cauchy problem and a forced IBVP with homogeneous data. A priori linear estimates for these decomposed problems are rigorously derived. (ii) The IBVP of the nonlinear KG equation is systematically analyzed via the Banach fixed‐point theorem in the space , which establishes local well‐posedness under the regularity condition , . (iii) A synthesis of the Fokas method with Sobolev spaces techniques extends the applicability of the Fokas method to fractional regularity regimes. The methodology provides explicit solution representations while maintaining appropriate regularity matching between initial and boundary data. This work significantly advances the functional framework for IBVP analysis on unbounded domains, bridging modern transform methods with classical Sobolev space theory.

Introduction Initial developments in computing were dominated by two classes of problems: (i) the jury or the boundary value problems – typically classified as elliptic partial differential equation in Chapter 3; (ii) the evolution or the initial–boundary value problems which are represented by parabolic and hyperbolic partial differential equations. In fact, the solution methods for heat equation (a parabolic partial differential equation) were central to the early development of the subject. These classical approaches are discussed in this chapter, with additional insight brought through spectral analysis of the schemes. It is noted that the stability analysis of numerical schemes was developed with respect to heat equation by von Neumann, as described in [41, 53]. This was considered a major milestone in the development of the subject. But, the readers' attention is also drawn to the correct analysis advanced recently, as described in [259] and Chapter 8, with respect to 1D convection equation. In fluid dynamics, a major milestone was the introduction of boundary layer concept by Ludwig Prandtl in 1904, which dominated fluid dynamics studies. Readers are referred to [209] for details of the development. Boundary layer equation is an example of parabolic partial differential equation.

A Feynman formula is a representation of the solution to the Cauchy problem for an evolution partial differential (or pseudodifferential) equation in terms of the limit of a sequence of multiple integrals with multiplicities tending to infinity. The integrands are products of the initial condition and Gaussian (or complex Gaussian) exponentials 1 [5]. In this paper, we obtain Feynman formulas for the solutions to the Cauchy problems for the Schrodinger equation and the heat equation with Levy Laplacian on the infinite-dimensional manifold of mappings from a closed real interval to a Riemannian manifold. The definition of the Levi Laplacian acting on functions on such a manifold is obtained by combining the methods of papers [3] and [7]. In the former, Levi Laplacians in the space of functions on an infinitedimensional vector space were considered, and in the latter, Volterra Laplacians in the space of functions on the above infinite-dimensional manifold were examined. This definition of a Levi Laplacian is equivalent to that given in [2], but it is better adapted for derivation of Feynman formulas. The main idea of the proof of the central result of this paper is reducing the derivation of Feynman formu1 For the heat equation, these multiple integrals coincide with integrals being finite-dimensional approximations to integrals with respect to the Wiener measure. For the Schrodinger equation, such integrals coincide with those used in the definition (which goes back to Feynman himself) of sequential Feynman path integrals. Therefore, the limits of multiple integrals in the Feynman formulas are integrals with respect to the Wiener measure in the former case and (sequential) Feynman path integrals in the latter case, and in both cases, the Feynman‐Kac-type formulas are consequences of the Feynman formulas being discussed. las for equations on a manifold to the derivation of similar formulas for equations on a vector space. For equations on finite-dimensional manifolds (containing the usual finite-dimensional Laplacians), this approach was suggested in [4] and developed in [6]. The Riemannian manifold under consideration was embedded in a suitable Euclidean space (this can always be done by the Nash theorem), and the technique of surface measures developed in [4, 7] was applied. An essential point in the proof was the application of the Chernoff formula (generalizing the Trotter formula), which is related to obtaining representations of solutions to evolution equations on manifolds (and to representations of solutions to equations on vector spaces in terms of path integrals in the phase space [5]) in the same way as the Trotter formula is related to representations of the solution to the simplest Schrodinger equation with potential in terms of path integrals in the configuration space. The remark made in the footnote means that the results obtained in this paper contain the construction

It is known that, if the time variable in the heat equation is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. The same is true for the Laplace equation on an infinite strip in the plane, regarded as an initial value boundary value problem. Also, it is known that if both variables, time and spatial, are complex, then, e.g. the Cauchy problem for the heat equation admits as solution, only a formal power series which, in general, converges nowhere. In a recent paper (C.G. Gal, S.G. Gal, and J.A. Goldstein, Evolution equations with real time variable and complex spatial variables, Complex Var. Elliptic Eqns. 53 (2008), pp. 753–774), a complementary approach was made: the study of the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ the spatial variable only (and keeping the time variable real). The goal of this article is to extend that study to the higher-order heat and Laplace-type equations. This ‘complexification’ is based on integral representations of the solutions in the case of a real spatial variable, by complexifying the spatial variable in the corresponding semigroups of operators. It is of interest to note that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness.