key: cord-0601455-qvb71k28
authors: Akel, Mohamed; Elshehabey, Hillal M.; Ahmed, Ragaa
title: Generalized integral transform method for solving multilayer diffusion problems
date: 2021-09-14
journal: nan
DOI: nan
sha: 20ccc8794744a5873cfc3728406065abf6d52a06
doc_id: 601455
cord_uid: qvb71k28

Multilayer diffusion problems have found significant important that they arise in many medical, environmental and industrial applications of heat and mass transfer. In this article, we study the solvability of one-dimensional nonhomogeneous multilayer diffusion problem. We use a new generalized integral transform, namely, $ bb M_{rho,m} -$transform [Srivastava et al., https://doi.org/10.1016/S0252-9602(15)30061-8]. First, we reduce the nonhomogeneous multilayer diffusion problem into a sequence of one-layer diffusion problems including time-varying given functions, followed by solving a general nonhomogeneous one-layer diffusion problem via the $ bb M_{rho,m} -$transform. Hence, by means of general interface conditions, a renewal equations' system is determined. Finally, the $ bb M_{rho,m} -$transform and its analytic inverse are used to obtain an explicit solution to the renewal equations' system. Our results generalize those ones in [Rodrigo and Worthy, http://dx.doi.org/10.1016/ j.jmaa.2016.06.042].

The multilayer diffusion problems are typical models for variety of solute transport phenomena in layered permeable media, such as advection, dispersion and reaction diffusions ( [13, 16, 19, 20, 23, 35] ). These problems have had their importance due to their natural prevalence in a remarkable large number of applications such as chamber-based gas fluxes measurements [22] , contamination and decontamination in permeable media [20, 21] , drug eluting stent [24, 28] , drug absorption [1, 32] , moisture propagation in woven fabric composites [27] , permeability of the skin [25] , and wool-washing [9] .

As epidemiological models, reaction-diffusion problems are widely used to model and analyze the spread of diseases such as the global COVID-19 pandemic caused by resulted from SARS-CoV2.

These models describe the spatiotemporal prevalence of the viral pandemic, and apprehend the dynamics depend on human habits and geographical features. The models estimate a qualitative harmony between the simulated prediction of the local spatiotemporal spread of a pandemic and the epidemiological collected datum. See [29, 36] . These data-driven emulations can essentially inform the respective authorities to purpose efficient pandemic-arresting measures and foresee the geographical distribution of vital medical resources. Moreover, such studies explore alternate scenarios for the repose of lock-down restrictions based on the local inhabitance densities and the qualitative dynamics of the infection. For more applications one can refer e. g., to [12, 15] .

Although the numerical methods are usually applied to solve the diffusion problems, especially in the heterogeneous permeable media, the analytic solutions when available, are characterized by their exactness and continuity in space and time. In this work, we focus on analytic solutions of certain nonhomgeneous diffusion problems in multilayer permeable media. Here, the retardation factors are assumed to be constant, the dispersion coefficients vary across layers, but being constants within each layer, and the free terms are (arbitrary) time-varying functions.

Analytic and semi analytic solutions of multilayer diffusion problems are developed by using the Laplace integral transform [6] , [5] , [7] , [8] , [11] , [14] , [15] , [20] , [30] , [34] , [37] . Applying Laplace transforms, to solve multilayer diffusion problems, has advantages as an applicable tool in handling different types of boundary conditions and averts solving complicated transcendental equations as in demand by eigenfunction expansion methods. Further works involving the Laplace transform have studied permeable layered reaction diffusion problem in [10] , [26] . Solutions obtained in these works are restricted to two layers as well as obtaining the inverse Laplace transform numerically.

In the current work, we aim to extend, generalize and merge results in [5] , [7] , [26] , [30] and [37] to solve certain nonhomgeneous diffusion problems in one-dimensional n-lyared media. We use a new generalized integral transform recently introduced in [33] . The obtained solutions are applicable to more general linear nonhomogeneous diffusion equations, finite media consisting of arbitrary many layers, continuity and dispersive flow at the contact interfaces between sequal layers and transitory boundary conditions of arbitrary type at the inlet and outlet. To the best knowledge of the authors, analytical solutions verifying all the above mentioned conditions have not previously reported in literature which strongly motivates this current work.

In the remaining part of this introductory section, in Subsection 1.1 the multilayer diffusion problem is described and then it is reformulated as a sequence of one-layer diffusion problems having boundary conditions including given time-depending functions. Basic properties for M ρ,m −transform that will be needed in this work are stated in Subsection 1.2. The remaining sections are constructed as follows: Section 2 is devoted to , we discuss the solvability of a gen-eral linear nonhomogeneous one-layer diffusion problem with arbitrary time-varying data, using the M ρ,m −transform. Section 3 is devoted to our main multilayer diffusion problem, where in Subsection 3.1 we solve a two-layer problem to shed light on the basic idea by considering this simple case. Further, in Subsection 3.2, we return to benfit from the results obtained in Section 2 and Subsection 3.1 to solve the main multilayer diffusion problem (1.1)-(1.7), see Subsection 1.1 below.

A one-dimensional diffusion problem in an n-layered permeable medium is set out as follows.

where d j 0, for all 1 j n, are the diffusion coefficients and

with m ∈ Z + = {1, 2, 3, · · · }, ρ ∈ C, Re(ρ) > 0. Here, the function-term λ(t, τ)r j (x, t) physically means the external source term that could be applied to the diffusion equation with r j (x, t) depends on time and space while the other factor of the source term i.e., p depends only on time. This last term could be for instance, a periodic-time magnetic source.

The initial conditions (ICs) are assumed as

The boundary conditions (BCs) are posited as • The outer BCs (at the inlet x = α and the outlet x = β) are general Robin boundary conditions as

for all t 0, with ı, ι, ℓ and l are constants satisfying, |ı| + |ι| > 0, |ℓ| + |l| > 0.

• The inner BCs (the interface conditions) are

for all t 0, with |ν j | + |µ j | > 0 for all j = 1, 2, · · · , n − 1.

For appropriate given functions η 1 , · · · , η n , ζ and ξ, we are going to find an analytic solution of the problem (1.1)-(1.7) using the M ρ,m −generalized integral transform, introduced recently in [33] .

Problem (1.1)-(1.7) can be reduced into the following sequence of one-layer diffusion problems.

• In the inlet layer i.e., x ∈ [x 0 , x 1 ]

(1.8)

• In the interior layers i.e., x ∈ [x j−1 , x j ], 2 j n − 1

(1.9)

• In the outlet layer i.e.,

ℓϕ n (x n , t) + l ∂ϕ n ∂x (x n , t) = λ(t, τ)ξ n (t), t 0, τ > 0. 

and

(1.13)

While, the outer boundary data ζ 1 (t) = ζ(t) and ξ n (t) = ξ(t) are given in (1.4) and (1.5), respectively, the functions ζ j (2 j n) can be determined once we specify the functions ξ j (1 j n − 1). Hence, we have to find ξ j , 1 j n − 1.

To do so, we should use the first matching condition (1.6).

In 

Setting ρ = 0 in (1.14), we recover the natural transform defined as (see [4] , [31] )

Thus, we have the following M ρ,m − N−transforms duality (1.17) and,

The Sumudu transform is defined by ( [2] , [3] and [17] )

Thus,

and,

Based on these dualities of the M ρ,m −transform (1.14) and these well-known integral transforms it seems to be interesting to apply the M ρ,m −transform (1.14) in solving a variety of boundary and initial-boundary problems. In this context, we recall the following results [33] :

• Let ϕ (n) (t) be the n th −order t−derivative of the function ϕ(t) and |ϕ(t)| Ke t/γ with

(1.20)

• Again, using the dualities stated before a convolution formula for the M ρ,m −transform (1.14)

can be obtained as follows. Here, the convolution for the Laplace transform will be considered, that is, for the functions ϕ and ψ, the convolution formula is given as

here, changing of the integral order is used. Thus, using the duality of the M ρ,m and N transforms (see (1.17)), we find

(1.21)

If we put ρ = 0 in (1.21), the case being interesting later in our work, then we

• Once again, using the dualities stated before an inversion formula of the M ρ,m −transform (1.14) is given (see, [33, Theorem 4 .1]) as 

The residue theorem (see e.g. [18] ) is usually used to calculate the contour integrals in (1.23) and (1.24).

Now, we investigate the solvability for the following one-layer nonhomogeneous initial-boundary value problem

where d, ı, ι, ℓ and l are constants such that |ı| + |ι| > 0, |ℓ| + |l| > 0, and p, r, η, ζ and ξ are given functions with p as in (1.2).

Applying the M ρ,m −transform defined by (1.14) to (2.1), yields 

then, (2.6) can be expressed aŝ

Applying the variation of parameters method to the nonhomogeneous equation (2.8), gives the general solution aŝ

where A and B are arbitrary invariants which can depend on s and τ.

Differentiating (2.10) with respect to x, giveŝ

Transforming the boundary conditions (2.3) and (2.4), implies

(2.12)

For simplicity, we set the following vector notations (2.14)

Substituting (2.10) and (2.11) into (2.12) and using the vector notation, give the algebraic linear

where · is the usual dot product in R 2 , and .15) is

is the determinant of the coefficient matrix of system (2.15). Substituting the constants A and B into (2.10) giveŝ

which can be rewritten aŝ

For further computation we rewriteθ(x; s, τ) aŝ

and

where ψ is given by (2.21) .

Proof. The first two conclusions of the lemma follow directly from the uniqueness theorem of initial value problem for second order ordinary differential equations having constant coefficients.

For fixed τ, s, y ∈ R and L ∈ R 2 , in view of (2.13) and (2.21) the functions L, C(y − x; s, τ) and

ψ(x, y; s, τ, L) are solutions to the following initial value problem It is easy to see that as functions in x both sides of (2.24) solve the differential equation in (2.26) and satisfy the initial conditions 

(2.28)

Next, in order to obtain the solution to the initial value problem ( 

where, The last integral can be usually calculated by the residue theorem [18] . Hence,

Res[e In view of (2.30) of Lemma 2.2 and (2.9), (2.28) can be rewritten aŝ

(2.39)

By the convolution formula (1.22), the inverse natural transform of (2.39) is 

The first conclusion is obvious when Θ 0 = 0 in (2.32). Thus, (2.41) can be simplified as (2.46)

Proof. The proof is similar to Lemma 2.2.

Hence, in view of the convolution formula (1.22) and the inversion of natural transform (1.24),

whereζ = λ(t, τ)ζ(t),ξ = λ(t, τ)ξ(t), δ 0 is the well-known Dirac delta function, and θ(x, t, τ) is given by (2.42) . Then, using the basic property of the Dirac delta function, that is δ 0 (ς)Φ(ς) = Φ(0), results in

Integrating by parts, gives

with θ(x, t, τ) is given by (2.42 ). This result can be rewritten as

where Γ k is the operator defined as 

When ρ = 0 and r j = ν j = 0, for all j = 1, · · · , n, Problem (2.1)-(2.4) and its

with Φ 0 (α − x; a), C(y, s k ) = C(y; s k , τ = 1) defined as (2.46), (2.13), respectively,

are reduced to that in [30, Section 3] .

Here, we are seeking the solution of our main problem defined in (1.1)-(1.7), which was converted into a sequence of initial boundary value problems (1.8)-(1.10). For the convenient of the reader and in order to draw the full picture in an easy way, we start with solving the bilayer diffusion problem in the following subsection, then we move to the general case in subsection 3.2.

For the two-layer problem, we have

and

Similar to what we denote in Section 2, we define the following vector notation a 1 = (ı, ι), b 1 = (ν 1 , µ 1 ), a 2 = (ν 2 , µ 2 ), b 2 = (ℓ, l), and Also, analogues to (2.19), define

(3.10)

Further, similar to (2.29), suppose that there are nonzero simple roots {s (1) k } ∞ k=1 and {s (2) k } ∞ k=1 of the functions ∆ 1 (s) and ∆ 2 (s), respectively. That is, 1, 2, ...) . (3.11) Therefore, according to (2.42), we obtain

k , τ) η 1 (y)dy (3.12)

k , τ) η 2 (y)dy, (3.13) where, Θ

0 and Θ (2) 0 can be defined as in Lemma 2.2. Also, similar to (2.48), with the respective forms Φ (1) 0 and Φ (2) 0 from Lemma 2.3 and the matching condition ξ 1 (t) = ζ 2 (t) we get

14)

k , τ)

where the operators Γ (1) k and Γ (2) k are obtained from (2.49). The matching condition ϕ 1 (x 1 , t) = Λ 1 ϕ 2 (x 1 , t) yields

For the unknown function ξ 1 we can rewrite the linear integral equation (3.16) as

where,

, (3.18) and

Inspire of the convolution formula (1.22) , the natural transform of (3.17) is

which can be rewritten as

That is,

for which the inverse natural transform is

where δ 0 is the Dirac delta function. Hence, we have

where for m 2, ψ m is the m−times self-convolution of ψ. Thus, one can conclude the solution to the bilayer diffusion problem (3.1)-(3.8) by the formulas (3.14) and (3.15) , together with (3.12) and (3.13) , with ξ 1 be given in (3.23) . Now, it's time to attack the main problem in the following subsection.

Here, we investigate the solvability of the main problem (1. Analogue to the computations of (2.42) and (2.48), we have for the current case, for all j = 1, · · · , n

k , τ) η j (y)dy, (3.29) where, Θ (j) 0 can be defined in a similar way as in Lemma 2.2, and 

in whichζ j = λ(t, τ)ζ j (t),ξ j = λ(t, τ)ξ j (t) and the linear operator T j is defined by (3.32) for all j = 1, · · · , n, L ∈ R 2 .

The matching conditions ϕ j (x j , t, τ) = Λ j ϕ j+1 (x j , t, τ), j = 1, ..., n − 1, lead to

In the sprite of the matching conditions (1.11), we haveζ j+1 (t) =ξ j (t) for all 1 j n − 1.

Thus, for j = 1, can be adjusted as a matrix equation

with A(t) is a tridiagonal matrix of order n − 1 whose entries: 

with the respective forms Φ (j) 0 and θ j defined as in (2.46) and (3.29) , respectively for all j = 1, ..., n.

Throughout the current contribution, a one-dimensional n-layer nonhomogeneous diffusion problem with time-varying data and general interface conditions have been concluded by means of a generalized integral transform. Although, most of the previous works have been focused on solving the problems of the homogeneous diffusion equation, the nonhomogeneous diffusion equation problem arises in many physical application. We have obtained the exact solutions for one-and multi-layer nonhomogeneous diffusion problems. The former case has been solved by a new generalized integral transform, the later one (n-layer problem) has been recast in a sequence of one layer problems. The obtained results generalize and extend those in [5] , [7] , [26] , [30] and [37] .

Our results motivate to deal with other types of diffusion problems. For example, reaction diffusion problems, Advection-reaction diffusion problems and non-autonomous reaction diffusion problems, etc.

On the other hand, more general partial differential equations (PDEs) and systems can be considered. for example, system of coupled PDEs, nonlinear diffusion PDEs and non-autonomous reaction diffusion PDEs. Those kinds of PDEs appear widely as epidemiological models to study and analyze the spread of diseases and pandemics [12, 15, 29, 36] .

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This project was supported by the Academy of Scientific Research and Tecchnology (ASRT), Egypt (Grant N0. 6407)