Higher-order heat and Laplace-type equations with real time variable and complex spatial variable

In two recent papers [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; C.G. Gal, S.G. Gal and J.A. Goldstein, Higher order heat and Laplace type equations with real time variable and complex spatial variable, Complex Var. Elliptic Eqns., 55 (2010), pp. 357–373, the classical heat and Laplace equations with real time variable and complex spatial variable are studied. The purpose of this article is to make a similar study for the classical wave and telegraph equations with real time variable and complex spatial variable. The complexification of the spatial variable in the wave and telegraph equations is made by two different methods which produce different equations. By the former method, we complexify the spatial variable in the corresponding formulas by replacing the usual translations x ± ct, c is the speed of propagation, by the rotations ze ±ict and, by the latter, we complexify the spatial variable in the corresponding evolution equation and then we search for analytic and non-analytic solutions. The first method produces solutions that also preserve some geometric properties of the boundary function, such as the univalence, starlikeness, convexity and spirallikeness. Moreover, new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated from both methods and their solutions are constructed. New physical/probabilistic interpretations of the solutions to these equations are also given.

It is known that if the time variable in the heat and wave equations is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. 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. The purpose of this article is, in a sense, complementary: to study the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ now the spatial variable only (and keeping the time variable real). This ‘complexification’ is studied by two different methods, which produce different equations: first, one complexifies the spatial variable in the corresponding semigroups of operators and secondly, one complexifies the spatial variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. It is of interest to note that in the case of the first method, besides the fact that the solutions can be studied by using the theory of semigroups of linear operators, also these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. Also, by both methods new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated and their solutions are constructed.

In a recent book co-authored by the authors of this article, we studied by semigroup theory methods several classical evolution equations, including the heat and Laplace equations, with real time variable and complex spatial variable, under the hypothesis that the boundary function belongs to the space of analytic functions in the open unit disk and continuous in the closed unit disk, endowed with the uniform norm. Also, in a subsequent paper, the authors have extended the results for the heat and Laplace equations in weighted Bergman spaces on the unit disk. The purpose of this article is to show that the semigroup theory methods work for these two evolution equations of complex spatial variables, under the hypothesis that the boundary function belongs to the weighted Fock space on \(\mathbb{C}\), \(F_{\alpha }^p(\mathbb{C})\), with \(1\leq p<+\infty \), endowed with The \(L^p\)-norm. Also, the case of several complex variables is considered. The proofs use the Jensen's inequality, Fubini's theorem for integrals and the \(L^p\)-integral modulus of continuity.
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This book investigates several classes of partial differential equations of real time variable and complex spatial variables, including the heat, Laplace, wave, telegraph, Burgers, Black-Merton-Scholes, Schroedinger and Korteweg-de Vries equations.The complexification of the spatial variable is done by two different methods. The first method is that of complexifying the spatial variable in the corresponding semigroups of operators. In this case, the solutions are studied within the context of the theory of semigroups of linear operators. It is also interesting to observe that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. The second method is that of complexifying the spatial variable directly in the corresponding evolution equation from the real case. More precisely, the real spatial variable is replaced by a complex spatial variable in the corresponding evolution equation and then analytic and non-analytic solutions are sought.For the first time in the book literature, we aim to give a comprehensive study of the most important evolution equations of real time variable and complex spatial variables. In some cases, potential physical interpretations are presented. The generality of the methods used allows the study of evolution equations of spatial variables in general domains of the complex plane.

In recent work, heat and Laplace equations, (un)damped wave equations, the Burgers and the Black–Merton–Scholes equations with real-time variable and complex spatial variable were studied. The purpose of this article is to make a similar study for the Schrödinger equation with real-time variable and complex spatial variable. The complexification of the spatial variable in the case of the Schrödinger equation is made by two different methods which produce different equations: first, one complexifies the spatial variable in the corresponding convolution formula by replacing the usual sum of variables (translation) by an exponential product (rotation) and second, one complexifies the spatial variable in the corresponding evolution equation and then one searches for non-analytic and for analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated and their solutions are constructed. It is of interest to note that in the case of the first method, solutions can be studied by employing the powerful theory of groups of linear operators. Then, we show that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. In the case of the higher order Schrödinger equation, the complexification of the spatial variable is made in the corresponding convolution formula.

The purpose of this article is to study the Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable. The complexification of the spatial variable in these equations is made by two different methods which produce different equations: first, one complexifies the spatial variable in the corresponding (real) solution by replacing the usual sum of variables (translation) by an exponential product (rotation) and secondly, one complexifies the spatial variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two dimensional spatial variables are generated and their solutions are constructed.

Higher-order heat and Laplace equations with complex spatial variables

The numerous examples of Chapters 2 and 3 demonstrate how the classical method of separation of variables is used to generate solutions of Laplace’s and the heat equation over specified finite domains. Utilizing the modern methods inherent in Green’s functions and Green’s Theorem, solutions of Poisson’s equation over an arbitrary finite domain are established. Can such modern methods be applied to the heat equation? Unlike Laplace’s or Poisson’s equations which are static in time, the heat equation evolves in time. Indeed, the heat equation is the prototype for what are commonly called evolution equations. Nonlinear evolution equations are examined in Chapter 6 and a special nonlinear system is detailed in Chapter 7. For now, the focus will be on the linear heat and wave equations. Instead of Green’s Theorem, one of the most powerful ideas in modern mathematics is applied: The Fourier transform. The program of study for this chapter then is to define the Fourier transform, develop its properties, apply it to the heat and wave equations, and derive analytic solutions.

This paper employs the weighted energy method to derive estimates for the dynamic behavior of solutions to boundary and initial boundary value problems with nonhomogeneous boundary conditions. In particular, the method is applied to the heat and Laplace equations in a bounded or unbounded region. Extensions to related equations are also studied. Similar estimates but for the spatial behavior is obtained for the heat equation and the backward in time heat equation. Results for blow-up in finite time of solutions to certain nonlinear equations are generalized to include nonhomogeneous boundary conditions, while solutions that vanish on part of the boundary are briefly discussed in the final section.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Nurlan Imanbaev, Nurzhan Erzhanov; Green's function of the heat equation with periodic and antiperiodic boundary conditions. AIP Conf. Proc. 16 December 2016; 1789 (1): 040027. https://doi.org/10.1063/1.4968480 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

The wave equation is the PDE $$ \frac{{\partial ^2 }} {{\partial t^2 }}u(x,t) - \Delta u(x,t) = 0 for x \in \Omega \subset \mathbb{R}^d , t \in (0,\infty ) or t \in \mathbb{R}. $$ ((6.1.1)) As with the heat equation, we consider t as time and x as a spatial variable. For illustration, we first consider the case where the spatial variable x is one-dimensional. We then write the wave equation as $$ u_{tt} (x,t) - u_{xx} (x,t) = 0. $$ ((6.1.2)) Let ϕ, ψ ∈ C2(ℝ). Then $$ u(x,t) = \phi (x + t) + \psi (x - t) $$ ((6.1.3)) obviously solves (6.1.2).KeywordsInitial DataWave EquationHeat EquationPropagation SpeedRepresentation FormulaThese 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.

In the present work, we construct solutions to a Fokker-Planck type equation with real time variable and complex spatial variable, and prove some properties.The equations are obtained from the complexification of the spatial variable by two different methods.Firstly, one complexifies the spatial variable in the corresponding convolution integral in the solution, by replacing the usual sum of variables (translation) by an exponential product (rotation).Secondly, one complexifies the spatial variable directly in the corresponding evolution equation and then one searches for analytic solutions.These methods are also applied to a linear evolution equation related to the Korteweg-de Vries equation. RESUMENEn este trabajo construimos soluciones de una ecuación tipo Fokker-Planck con variable de tiempo real y variable espacial compleja.Las ecuaciones se obtienen de la complejización de la variable espacial por dos métodos diferentes.Primero, se complejiza la variable espacial en la integral de convolución respectiva en la solución reemplazando la suma usual de las variables (traslaciones) por un producto de exponenciales (rotación).Luego, se complejiza la variable espacial directamente en la respectiva la ecuación de evolución y se busca por las soluciones analíticas.Estos métodos también se aplican a la ecuación de evolución lineal relacionada a la ecuación Korteweg-de Vries.

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.

Partial differential equations (PDEs) are used in the real world to model physical phe- nomena such as heat, wave, Laplace, and Poisson equations. For regular shape domains, the heat equation can be solved analytically; however, for irregular domains, the computation of the solu- tion is difficult and numerical methods like Finite Difference Method (FDM) and Finite Element Method (FEM) can be used. FEM provides approximate values at discrete points in the domain. It breaks down a large problem into smaller finite elements. These element’s equations are combined into a system representing the whole problem. We show the comparison between analytic solution, solutions by FDM and FEM. The impact of heat on the material is examined at various positions and multiple positions. We compare the analytical and numerical (by FEM) solution considering several homogeneous materials with various diffusivity values (α). Finally, the simulation results of different non-homogeneous materials were compared. Science and engineering fields that use heat equations can be evaluated using the numerical method applied here.

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