1. Introduction

Partial differential equations (PDEs) are more difficult to solve than ordinary differential equations (ODEs). Therefore, numerical approximations are widely used in application. There are standard numerical methods such as finite difference and finite element etc. to solve (1). In numerical approaches to solving PDEs, particularly in finite difference methods, we may face two ways of accuracy loss because of discretization of domain and approximating partial derivatives by difference formulas. Finite element methods will improve the accuracy of problems with irregular boundaries, but still abide by approximations as one has to do minimizing functionals with smaller class of functions [1]. Also, artificial neural network methods can be used for approximate nonsmooth solutions. See [2] for application of artificial neural network methods for Poisson’s equation with boundary value problems in domain ℝ². This method trains data to optimize the algorithm which gives numerical solutions to partial differential equations. Convergent properties and accuracy of solution are not discussed in the given method of [2]. Finite element method can be used to solve elliptic problems with Henstock–Kurzweil integrable functions. Such an approximation is used in [3] for ODEs, where existence and uniqueness of the solution are not discussed. The proposed analytical approach in our paper will direct to a solution, where if its explicit form is not available, one has to only do a numerical integration.

There are several methods to find analytical solution for Dirichlet problem such as Green’s function, Dirichlet’s principle, layer potentials, energy methods, and Perron’s method. In 1850 [4], George Green gives theoretical approach for Dirichlet problem which is now called as Green’s function. This assumes that Green’s function exists for any domain which is not true. However, this idea influences modern techniques and distributional theory. Using Green’s function, one can obtain a general representation solution for Poisson’s equation.

General representation solution depends on the domain Ω and nature of the functions f and g. For Lebesgue integration, it is required that the conditions g ∈ C(∂Ω) and f ∈ C(Ω) get a unique analytical solution $$ for (1), where Ω needs to be open and bounded in ℝⁿ ([5], pg. 28). In classical solution, it is worth to use Henstock–Kurzweil integral (HK-integral) instead of the Lebesgue integral.

HK-integral is more advanced, and it includes the Lebesgue and Riemann integral. This integral was first introduced by Henstock and Kurzweil in 1957. It has many advantages such as convergence theorems, integration on unbounded intervals/functions, Fubini’s theorem, and fundamental theorem of calculus with full generality, for details see [6–9]. If $$ is the space of HK-integrable functions, then it has proper inclusion $$. For an example, a highly oscillating function f(x) = (1/x)sin(1/x³) is neither Lebesgue nor Riemann integrable. But f is HK-integrable. There are only few articles which use HK-integration to find analytical solution for Dirichlet problem [10, 11]. Talvila uses HK-integration to find solution for Dirichlet problem in the upper half plane, see example in [10]. Also, the same author obtains useful results of Poisson kernel in the unit disk via HK-integral with application into Dirichlet problem [11]. However, both articles [10, 11] prove its results in ℝ² and do not extend into ℝ³. It is an obvious fact that properties of HK-integration are not easily extended for higher dimension. Even the HK-integration is superior to Lebesgue integration, there are a few drawbacks. Most significantly, there is no known natural topology for $$, so that it is not a Banach space. Therefore, there are not much functional analysis tools to work with the problem. However, there is a natural seminorm on $$ defined by Alexiewicz [12].

Looking at an alternative, we moved into distribution theory; specially the space of integrable distribution $$ which is isomorphic to completion of the space $$, which is presented in Section 2. It seems the concept of integrable distribution is first introduced by Mikusiński and Ostaszewski [13] and further developed by several authors [14, 15]. In recent years, integrable distributional solution for differential equations has been used by many authors [16–19]. We define derivative (in distributional sense) of $$ and the integral on it. In Section 3, the space of Hardy–Krause variation functions is identified as a multiplier for $$. Here it uses Henstock–Stieltjes integral and integration by parts defined on $$. In [15], example 14.4, Talvila discussed the Dirichlet boundary condition of Laplace equation in the half space. There it uses Poisson Kernel, and proof of integrable distributional solution for the Laplace equation is not given. We extend the proof of this problem in Section 4 and obtain integrable distributional solution for (1) when f = 0.

2. Integrable Distributions

Distribution theory was frequently used with the advent of Laurent Schwartz [20] in 1950. Distributions or generalized functions are an important functional analysis tool in modern analysis, especially in the field of PDE’s. When functions are nonsmooth, then distribution theory allows to perform operations: translation, differentiation, convolution etc. Here, we use integrable distributions which are a special class of distributions that will be important for applications of distribution theory to partial differential equations, for details see [13, 14, 21]. There are several articles on solving ordinary and partial differential equations using integrable distributions [16, 17, 19]. Integrable distribution has used to find a distributional solution for Poisson’s equation with Dirichlet boundary condition for the upper half plane. This article extends the distributional solution for Poisson’s equation for the 3-dimensional upper half space.

Let the space of test functions is $$ and ϕ has compact support}. Then, $$ forms a normed space under usual pointwise operations. The dual space, $$, is called as the space of distributions. That is, the distributions are the continuous linear functionals on $$. For a given distribution T, denotes its action on test function $$ by 〈T, ϕ〉. For an example, if f is a locally HK-integrable function, then its corresponding distribution is defined $$. Hereafter, all integral will be HK-integral unless stated otherwise. Next, consider derivative of a distribution, sometimes called as distributional derivative or weak derivative. If F is continuously differentiable, then $$ for any test function ϕ. The compact support of ϕ was used in this integration by parts. With this suggestion, we define the distributional derivative of a distribution $$ by 〈∂^αF, ϕ〉 = (−1)^|α|〈F, ∂^αϕ〉 for any $$. Therefore, in particular, 〈∂₁₂F, ϕ〉 = 〈F, ∂₁₂ϕ〉, where ∂₁₂ = ∂²/∂y ∂x. Now, consider a specific type of distributions called as integrable distributions. This definition seems to have been first introduced by Mikusiński and Ostaszewski [13]. After that, it was developed in detail in the plane by Ang et al. [14] and on $$ and $$ by Talvila [15, 21].

For the extended real numbers $$, consider $$ be the space of continuous functions so that both lim_x⟶−∞F(x, y) and lim_y⟶−∞F(x, y) exit for any $$. Then, define $$. Then, $$ forms a Banach space under the uniform norm ‖⋅‖_∞. Next, define the integrable distribution as the derivative of function in $$.

Here, f = ∂₁₂F be the sense of distributional derivative, i.e., 〈f, ϕ〉 = 〈∂₁₂F, ϕ〉 = 〈F, ∂₁₂ϕ〉 for $$. The function F is called as a primitive of f. Generally, primitive is not unique and differed by a constant. However, with our choice F(−∞, y) = F(x, −∞) = 0, primitive is unique for any given $$.

Note that the space $$ with the Alexiewicz norm is a normed space, but not a Banach space [12, 22]. Absence of Banach space structure of $$ is main disadvantage on it. It is one of the reasons to move into $$.

Alexiewicz norm dose not induce an inner product in $$ because the norm does not satisfy the parallelogram law. Therefore, $$ is not a Hilbert space under the norm ‖⋅‖. However, the space $$ with ‖⋅‖ is isomorphic to $$ with ‖⋅‖_∞. The mapping f⟶F gives a linear isomorphism between the spaces. Uniqueness of primitive gives the mapping is injective, and by the definition it is surjective mapping. Furthermore, the mapping preserves norm since ‖f‖ = ‖F‖_∞. Thus, the space $$ is identified with the space of continuous functions vanish at infinity.

Next, we define the integral of an integrable distribution, as it uses the primitive $$ of $$.

This integral is uniquely defined by uniqueness of the primitive, Theorem 1. Also, it is very general and includes the Riemann integral, Lebesgue integral, and HK-integral. Norm preservation of Proposition 1 gives that the corresponding integral values agree on each integral as presented in Theorem 3.

Note that if f ∈ L¹(ℝ²), then its Alexiewicz norm and L¹ norm are not equivalent, for an example see [15] (pg. 9). The norms are equivalent if f ≥ 0 for almost everywhere. From the definition of integral of $$ and its unique primitive, we obtain following fundamental theorem of calculus.

3. Multiplier for Integrable Distributions

We will turn now to our discussion on multipliers for $$. A multiplier for a class of functions ℱ is a function g such that fg ∈ ℱ for each f ∈ ℱ. For an example, in Lebesgue integral, g ∈ L^∞(ℝⁿ) if and only if fg ∈ L¹(ℝⁿ) for each f ∈ L¹(ℝⁿ). i.e., the space L^∞(ℝⁿ) is multipliers for L¹(ℝⁿ). For HK-integral in one-dimensional case 2, space of bounded variation functions, $$, is multipliers for $$ [9], (Theorem 6.1.5). In multidimension case, there is no unique way to extend the notion of variation to function. Adams and Clarkson [23] give six such extensions but mostly recognized by Vitali variation and Hardy–Krause variation. To establish integration by parts formula and multipliers for $$, we begin with Hardy–Krause variation.

The set of functions in Hardy–Krause variation is denoted by $$. As in [15], $$ forms a Banach space with norm defined by $$. Here, V₁φ and V₂φ are bounded variation to the functions of (ii) and (iii) in Definition 3. For any multidimensional case, any function of Hardy–Krause variation is a multiplier for HK-integrable functions, see [9]. Therefore, define following integration by parts, as a two-dimensional case which is presented in ([9], Theorem 6.5.9).

Here, dg(x, y) is Henstock–Stieltjes integral of relevant variables, see in detail [24]. This integration by parts formula induces the space of Hardy–Krause bounded variation functions, $$, as a multiplier for $$. This multiplier is important to define convolution and obtain convergence theorem.

It needs to obtain continuity of the Alexiewicz norm since the Dirichlet boundary condition is taken on it. For any (s, t) ∈ ℝ², the translation of $$ is defined by 〈τ_{(s, t)}f, ϕ〉 = 〈f, τ_{(−s, −t)}ϕ〉 for $$, where τ_{(s, t)}ϕ(x, y) = ϕ(x − s, y − t). Translation is invariant and continuous under the Alexiewicz norm.

4. Integrable Distributional Solution for Poisson Equation

Convergence theorems on Banach space are important when solving partial differential equations. There are several convergence theorems in $$ such as strong convergence in $$, weak/strong convergence in $$, and weak/strong convergence in $$. Details are discussed in the plane by Ang, Schmitt and Vy [14] and on $$ by Talvila [2]. Now, we move into the convergence theorem on $$ which will help to interchange limit of integral.

This convolution gives integrable distributional solution for (16).

Next, we give an example which will apply Theorem 6.

Even HK-integral exists in (27), it may not be able to evaluate and get explicit function. In that case, numerical integration can be applied. Numerical integration should be done with respect to s and t while fixing x, y, and z, details in [26]. This reduces solving the partial differential equation into applying numerical integration for the double integral (27).

5. Discussion

In this work, integrable distributional solution is obtained for Poisson equation with the Dirichlet boundary condition in the upper half space. Boundary conditions are taken in the Alexiewicz norm. Lebesgue integral uses for the classical solution to Poisson equation. Because of the proper inclusion L¹⊊  ℋR, we may use HK-integral to solve Poisson equation. However, there are some limitations due to the absence of any natural topology. That is, $$ is not a Banach space; nevertheless, there is a natural seminorm called Alexiewicz. This is one of the main reasons to move into the space of integrable distribution $$, which contains $$ and leads to a Banach space with Alexiewicz norm.

Integrable distribution is defined as derivative (distributional sense) of function in $$. In this definition, any $$ identifies with the unique primitive $$. This unique primitive allows to define a Alexiewicz norm ‖⋅‖ on $$ and leads to a Banach space. Alexiewicz norm does not induce an inner product in $$ because it does not satisfy the parallelogram law. Therefore, $$ is not a Hilbert space under the norm ‖⋅‖. However, the space $$ is isometrically isomorphic to the space $$. Integral on $$ is uniquely defined by uniqueness of the primitive. $$ is the completion of L¹(ℝ²) (or $$) with respect to the corresponding Alexiewicz norm.

In one-dimensional HK-integral, space of bounded variation functions, $$, is multipliers for $$. In multidimension case, there are several extensions to variation of function. To establish integration by parts formula and multipliers for $$, we begin with Hardy–Krause variation. The space of Hardy–Krause bounded variation functions, $$, as a multiplier for $$. This multiplier is important to define convolution and obtain convergence theorem. $$ is used to define convolution and obtain convergence theorem on $$. There are several convergence theorems in $$ such as strong convergence in $$, weak/strong convergence in $$, and weak/strong convergence in $$. Convergence theorem on $$ and dominated convergence theorem in $$ are used to obtain integrable distributional solution. Finding of this article gives solution for Poisson equation for broader initial condition in the upper half space. Dirichlet problem in ℝ³ is considered, and the solution has obtained by HK-integration where it is not possible from Lebesgue integral. Because of function’s complexity, it may not be able to have solution in explicit form. It would help keep an analytical outlook in the solution, which would be better than solving the partial differential equation by numerical methods such as finite difference methods and finite element methods.

There is potential to find integrable distributional solution for Poisson equation in other boundary conditions, Neumann and Robin. But it needs to develop suitable convergence theorems on $$ and its multipliers $$. Also, it is possible to consider Poisson equation in unit ball with the relevant Poisson kernel. Talvila, in [15], gives some construction to develop integrable distribution on ℝⁿ, integration by parts, and mean value theorem and defines $$ and $$ for multidimensional case. Hence, integrable distributional solution is possible to obtain for Poisson equation with Dirichlet boundary condition in $$ with suitable convergence theorem.

Conflicts of Interest

The authors declare that they have no conflicts of interest.