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Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations

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Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2021, Article ID 5511283, 9 pages https://doi.org/10.1155/2021/5511283 Research Article Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations Amila J. Maldeniya ,1 Naleen C. Ganegoda ,2 Kaushika De Silva,2 and Sanath K. Boralugoda3 1Department of Mathematics, General Sir John Kotelawala Defence University, Kandawala Road, Ratmalana, Sri Lanka 2Department of Mathematics, University of Sri Jayewardenepura, Gangodawila, Nugegoda, Sri Lanka 3Department of Mathematics, University of Colombo, Cumarathunga Munidasa Mawatha, Colombo, Sri Lanka Correspondence should be addressed to Amila J. Maldeniya; [email protected] Received 11 February 2021; Revised 14 May 2021; Accepted 24 June 2021; Published 12 July 2021 Academic Editor: Ra´ul E. Curto Copyright © 2021 Amila J. Maldeniya et al. &is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we present some properties of integrable distributions which are continuous linear functional on the space of test function D(R2). Here, it uses two-dimensional Henstock–Kurzweil integral. We discuss integrable distributional solution for 3 Poisson’s equation in the upper half space R+ with Dirichlet boundary condition. 1. Introduction improve the accuracy of problems with irregular boundaries, but still abide by approximations as one has to do mini- In this paper, we find an integrable distributional solution mizing functionals with smaller class of functions [1]. Also, for Poisson’s equation in the upper half space. Poisson’s artificial neural network methods can be used for approx- equation is a second-order elliptic partial differential imate nonsmooth solutions. See [2] for application of ar- equation which has many applications in physics and en- tificial neural network methods for Poisson’s equation with gineering. &ere are three types of boundary conditions for boundary value problems in domain R2. &is method trains Poisson’s equation: Dirichlet condition, Neumann condi- data to optimize the algorithm which gives numerical so- tions, and Robin conditions. &is article is mainly con- lutions to partial differential equations. Convergent prop- cerning on Poisson’s equation with Dirichlet boundary erties and accuracy of solution are not discussed in the given condition (1): method of [2]. Finite element method can be used to solve n elliptic problems with Henstock–Kurzweil integrable △u � f; in Ω ⊂ R ; functions. Such an approximation is used in [3] for ODEs, ( (1) u � g; on zΩ : where existence and uniqueness of the solution are not discussed. &e proposed analytical approach in our paper Partial differential equations (PDEs) are more difficult to will direct to a solution, where if its explicit form is not solve than ordinary differential equations (ODEs). &ere- available, one has to only do a numerical integration. fore, numerical approximations are widely used in appli- &ere are several methods to find analytical solution for cation. &ere are standard numerical methods such as finite Dirichlet problem such as Green’s function, Dirichlet’s difference and finite element etc. to solve (1). In numerical principle, layer potentials, energy methods, and Perron’s approaches to solving PDEs, particularly in finite difference method. In 1850 [4], George Green gives theoretical ap- methods, we may face two ways of accuracy loss because of proach for Dirichlet problem which is now called as Green’s discretization of domain and approximating partial deriv- function. &is assumes that Green’s function exists for any atives by difference formulas. Finite element methods will domain which is not true. However, this idea influences 2 International Journal of Mathematics and Mathematical Sciences modern techniques and distributional theory. Using Green’s equation is not given. We extend the proof of this problem in function, one can obtain a general representation solution Section 4 and obtain integrable distributional solution for (1) for Poisson’s equation. when f � 0. General representation solution depends on the domain Ω and nature of the functions f and g. For Lebesgue in- 2. Integrable Distributions tegration, it is required that the conditions g ∈ C(zΩ) and f ∈ C(Ω) get a unique analytical solution Distribution theory was frequently used with the advent of u ∈ C2(Ω) ∩ C(Ω) for (1), where Ω needs to be open and Laurent Schwartz [20] in 1950. Distributions or generalized bounded in Rn ([5], pg. 28). In classical solution, it is worth functions are an important functional analysis tool in to use Henstock–Kurzweil integral (HK-integral) instead of modern analysis, especially in the field of PDE’s. When the Lebesgue integral. functions are nonsmooth, then distribution theory allows to HK-integral is more advanced, and it includes the perform operations: translation, differentiation, convolution Lebesgue and Riemann integral. &is integral was first in- etc. Here, we use integrable distributions which are a special troduced by Henstock and Kurzweil in 1957. It has many class of distributions that will be important for applications advantages such as convergence theorems, integration on of distribution theory to partial differential equations, for unbounded intervals/functions, Fubini’s theorem, and details see [13, 14, 21]. &ere are several articles on solving fundamental theorem of calculus with full generality, for ordinary and partial differential equations using integrable details see [6–9]. If HK is the space of HK-integrable distributions [16, 17, 19]. Integrable distribution has used to functions, then it has proper inclusion L1 ⊊ HK. For an find a distributional solution for Poisson’s equation with example, a highly oscillating function f(x) � (1/x) Dirichlet boundary condition for the upper half plane. &is sin(1/x3) is neither Lebesgue nor Riemann integrable. But f article extends the distributional solution for Poisson’s is HK-integrable. &ere are only few articles which use HK- equation for the 3-dimensional upper half space. integration to find analytical solution for Dirichlet problem Let the space of test functions is [10, 11]. Talvila uses HK-integration to find solution for D(R2) � �ϕ: R2 ⟶ Rjϕ ∈ C∞ and ϕ has compact sup- Dirichlet problem in the upper half plane, see example in port}. &en, D(R2) forms a normed space under usual [10]. Also, the same author obtains useful results of Poisson pointwise operations. &e dual space, D′(R), is called as the kernel in the unit disk via HK-integral with application into space of distributions. &at is, the distributions are the Dirichlet problem [11]. However, both articles [10, 11] prove continuous linear functionals on D(R2). For a given dis- its results in R2 and do not extend into R3. It is an obvious tribution T, denotes its action on test function ϕ ∈ D(R2) by fact that properties of HK-integration are not easily ex- hT; ϕi. For an example, if f is a locally HK-integrable tended for higher dimension. Even the HK-integration is function, then its corresponding distribution is defined hT ; i � R∞ R∞ f superior to Lebesgue integration, there are a few drawbacks. f ϕ − ∞ − ∞ ϕ. Hereafter, all integral will be HK- Most significantly, there is no known natural topology for integral unless stated otherwise. Next, consider derivative of HK, so that it is not a Banach space. &erefore, there are not a distribution, sometimes called as distributional derivative or weak derivative. If F is continuously differentiable, then much functional analysis tools to work with the problem. ∞ ∞ R F′ � − R F ′ However, there is a natural seminorm on HK defined by − ∞ ϕ − ∞ ϕ for any test function ϕ. &e compact Alexiewicz [12]. support of ϕ was used in this integration by parts. With this Looking at an alternative, we moved into distribution suggestion, we define the distributional derivative of a F ′( ) hzαF; i � (− )jαjhF; zα i theory; specially the space of integrable distribution AC distribution ∈ D R by ϕ 1 ϕ for which is isomorphic to completion of the space HK, which any ϕ ∈ D(R2). &erefore, in particular, 2 is presented in Section 2. It seems the concept of integrable hz12F; ϕi � hF; z12ϕi, where z12 � z /zy zx. Now, consider distribution is first introduced by Mikusinski´ and Ostas- a specific type of distributions called as integrable distri- zewski [13] and further developed by several authors butions. &is definition seems to have been first introduced [14, 15]. In recent years, integrable distributional solution for by Mikusinski´ and Ostaszewski [13]. After that, it was de- veloped in detail in the plane by Ang et al. [14] and on R and differential equations has been used by many authors 2 A R by Talvila [15, 21]. [16–19]. We define derivative (in distributional sense) of C 2 and the integral on it. In Section 3, the space of Har- For the extended real numbers R, consider C0(R ) be dy–Krause variation functions is identified as a multiplier for the space of continuous functions so that both HK. Here it uses Henstock–Stieltjes integral and integra- limx⟶− ∞F(x; y) and limy⟶− ∞F(x; y) exit for any A 2 2 tion by parts defined on C. In [15], example 14.4, Talvila F ∈ C(R ). &en, define B (R 2) � �F ∈ C0(R ) jF(− ∞; discussed the Dirichlet boundary condition of Laplace C 2 equation in the half space. &ere it uses Poisson Kernel, and y) � F(x; − ∞) � 0; ∀x; y ∈ Rg. &en, BC(R ) forms a proof of integrable distributional solution for the Laplace Banach space under the uniform norm k·k∞. Next, define the International Journal of Mathematics and Mathematical Sciences 3 integrable distribution as the derivative of function in (i) If f ≡ 0, then clearly kfk � 0. If kfk � 0 and then 2 kFk � 0. &is implies F ≡ 0 and hence BC(R ).
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