Generalized Laplace-Type Transform Method for Solving Multilayer Diffusion Problems

Akel, Mohamed; Elshehabey, Hillal M.; Ahmed, Ragaa

doi:https://doi.org/10.1155/2022/2304219

Journal of Function Spaces

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 2304219 | https://doi.org/10.1155/2022/2304219

Generalized Laplace-Type Transform Method for Solving Multilayer Diffusion Problems

Mohamed Akel,^1,2Hillal M. Elshehabey,^1,2and Ragaa Ahmed^1,2

Academic Editor: Lars E. Persson

Received08 Oct 2021

Accepted27 Nov 2021

Published07 Mar 2022

Abstract

Multilayer diffusion problems have found significant importance that they arise in many medical, environmental, and industrial applications of heat and mass transfer. In this article, we study the solvability of a one-dimensional nonhomogeneous multilayer diffusion problem. A new generalized Laplace-type integral transform is used, namely, the -transform. 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 -transform. Hence, by means of general interface conditions, a renewal equations’ system is determined. Finally, the -transform and its analytic inverse are used to obtain an explicit solution to the renewal equations’ system. Our results are of general attractiveness and comprise a number of previous works as special cases.

1. Introduction

The multilayer diffusion problems are typical models for a variety of solute transport phenomena in layered permeable media, such as advection, dispersion, and reaction diffusions [1–10]. These problems have had their importance due to their natural prevalence in a remarkably large number of applications such as chamber-based gas flux measurements [11], contamination and decontamination in permeable media [6, 12], drug-eluting stent [13, 14], drug absorption [15, 16], moisture propagation in woven fabric composites [17], permeability of the skin [18], and wool-washing [19]. Further applications have been considered in [20, 21].

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 SARS-CoV-2. These models describe the spatiotemporal prevalence of the viral pandemic and apprehend the dynamics depending 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 [22, 23]). 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 lockdown restrictions based on the local inhabitance densities and the qualitative dynamics of the infection. For more applications, one can refer, e.g., to [24, 25].

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 the context of obtaining numerical solutions for such models, we refer to the following references [26–30]. In this work, we focus on analytic solutions of certain nonhomogeneous 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 semianalytic solutions of multilayer diffusion problems are developed by using the integral transforms [6, 25, 31–40]. 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 demanded by eigenfunction expansion methods. Further works involving the Laplace transform have studied the permeable layered reaction diffusion problem in [41, 42]. Solutions obtained in these works are restricted to two layers as well as obtaining the inverse Laplace transform numerically. In the same context, generalized integral transform techniques, for short GITT, are well-established hybrid approaches for solving diffusion and convection-diffusion problems, in which hybrid refers to the combination of classical analytical methods with modern computational tools aimed at accurate, robust, and low-cost solutions [43–47]. In the current work, we aim to extend, generalize, and merge results in [31, 33, 38, 40, 42] to solve certain nonhomogeneous diffusion problems in one-dimensional -layered media. We use a new generalized integral transform recently introduced in [48]. 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 the arbitrary type at the inlet and outlet. To the best knowledge of the authors, analytical solutions verifying all the above mentioned conditions have not been previously reported in the 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 the-transform that will be needed in this work are stated in Subsection 1.2. The remaining sections are constructed as follows: in Section 2, we discuss the solvability of a general linear nonhomogeneous one-layer diffusion problem with arbitrary time-varying data, using the -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 benefit from the results obtained in Section 2 and Subsection 3.1 to solve the main multilayer diffusion problems (2)–(8) (see Subsection 1.1 below).

1.1. Mathematical Modeling for Nonhomogeneous -Layer Diffusion Systems

A one-dimensional diffusion problem in an -layered permeable medium is set out as follows. Let be a finite partition of the interval . In each subinterval , with , the component function satisfies the partial differential equation (PDE) where , for all , are the diffusion coefficients and with . Here, the function-term physically means the external source term that could be applied to the diffusion equation with depends on time and space while the other factor of the source term, i.e., , depends only on time. This last factor could be, for instance, a periodic-time magnetic source.

The initial conditions (ICs) are assumed as

The boundary conditions (BCs) are posited as (i)The outer BCs (at the inlet and the outlet ) are general Robin boundary conditions as

for all , with , and are constants satisfying . (ii)The inner BCs (the interface conditions) arefor all , with for all .

For appropriate given functions , and , we are going to find an analytic solution of the problems (2)–(8) using the -generalized integral transform, introduced recently in [48]. Problems (2)–(8) can be reduced into the following sequence of one-layer diffusion problems. (i)In the inlet layer, i.e., ,(ii)In the interior layers, i.e., ,(iii)In the outlet layer, i.e., ,

Remark 1. Each of the initial boundary value problems (9)–(11) is a case of the one-layer nonhomogeneous diffusion problem that will be discussed in Section 2 below.

Now, in view of the inner boundary conditions (7) and (8), the time-varying functions and for all are subject to so that

While the outer boundary data and are given in (5) and (6), respectively, the functions can be determined once we specify the functions . Hence, we have to find . To do so, we should use the first matching condition (7).

1.2. Srivastava-Luo-Raina Generalized Integral Transform

In [48], Srivastava et al. introduced the following generalized integral transform: for a continuous (or piecewise continuous) function on , where is the transform variable and is a parameter. The basic properties of the -transform are given in [48]. Next, we recall some of these properties, which are needed in the present work. Indeed, as introduced in [48] the -transform is closely related with the well-known integral transforms, the Laplace, natural, and Sumudu transforms. The Laplace transform is defined by

So, from (14) and (15), we have the following duality relations:

Setting in (14), we recover the natural transform defined as (see [49, 50])

Thus, we have the following -transform duality

The Sumudu transform is defined by [51–53]

Thus,

Based on these dualities of the -transform (14) and these well-known integral transforms, it seems to be interesting to apply the -transform (14) in solving a variety of boundary and initial-boundary value problems. In this context, we recall the following results [48]: (i)Let be the -order -derivative of the function and with , . Then,where is defined by (18). Using the duality (21) in (25), we find (ii)Again, using the dualities stated before a convolution formula for the -transform (14) can be obtained as follows. Here, the convolution for the Laplace transform will be considered; that is, for the functions and , the convolution formula is given as

If and , thenwhere

Setting in the last equality, one gets Here, changing of the integral order is used. Thus, using the duality of the and transforms (see (20)), we find

Remark 2. If we put in (30), the case being interesting later in our work, then we get (iii)Once again, ...again, using the dualities stated before an inversion formula of the -transform (14) is given (see Theorem 4.1 of [48]) asas long as the integral converges absolutely. In case, when one obtains the following inversion formula of the natural transform (see Theorem 5.3 of [49])

The residue theorem (see, e.g., [54]) is usually used to calculate the contour integrals in (32) and (33).

2. One-Layer Nonhomogeneous Diffusion System

Now, we investigate the solvability for the following one-layer nonhomogeneous initial boundary value problem: where , and are constants such that , and , and are given functions with as in (3).

Applying the -transform defined by (14), to (34), yields

Using the duality of the -transform and the natural transform given by (21) and (25), Equation (38) can be reduced to where is defined by (18). Setting then, (39) can be expressed as where

Applying the variation of the parameter method to the nonhomogeneous equation (41) gives the general solution as where and are arbitrary invariants which can depend on and . Differentiating (43) with respect to , gives

Transforming the boundary conditions (36) and (37), implies

For simplicity, we set the following vector notations:

Obviously, we have

Substituting (43) and (44) into (45) and using the vector notation, we give the algebraic linear system where is the usual dot product in , and with and and are the boundary data given in (36) and (37), respectively. The solution of system (50) is where is the determinant of the coefficient matrix of system (50). Substituting the constants and into (43) gives which can be rewritten as where

For further computation, we rewrite as

Lemma 3. Let and . Then, Consequently, for each zero of the function , one has where is given by (57).

Proof. The first two conclusions of the lemma follow directly from the uniqueness theorem of the initial value problem for the second-order ordinary differential equations having constant coefficients.
For fixed and , in view of (46) and (57), the functions and are solutions to the following initial value problem: Thus, with the uniqueness of the solution to problem (63), we conclude (60).
It is easy to see that as functions in , both sides of (61) are linear combinations of the functions which are linearly independent solutions to the differential equation in (63). Thus, both sides of (61) solve this differential equation.
Moreover, in view of (57), both sides of (61) satisfy Hence, by the uniqueness theorem, (61) holds true.
For each being a zero of the function , taking the limit in both sides of (61) as gives (62).
Applying Lemma 3, (56) and (59) respectively can be reduced to Next, in order to obtain the solution to the initial value problem (34)–(37), we apply the inversion formula (33) to (65) and (66). In doing so, we suppose that there are nonzero simple roots of . That is,

Lemma 4. Suppose that (67) holds true. For each and , we get where with

Proof. Let Applying the inversion formula (33), we find The last integral can be usually calculated by the residue theorem [54]. Hence, Recalling (67), each is a simple pole of . Therefore, At , we have We see that either as tends to . Then, is either a removable singular point or a simple pole of .
Hence, substituting (74) and (75) in (73) gives the main conclusion of the Lemma, i.e., (69) and (70).☐

In view of (68) and (42), we have where is defined by (69) and (70), and

Hence, (66) can be rewritten as

By the convolution formula (31), the inverse natural transform of (79) is

That is,

From Lemma 3, one has at and the zeros of . That results in

The first conclusion is obvious when in (70). Thus, (81) can be simplified as

Next, we return to (65). Using (26) (for ) and (51), (65) can be rewritten as where is given in (66). Now, we can obtain the solution of Problem (34)–(37) by operating the inversion formula (33) in (85). In doing so, we need the following lemma.

Lemma 5. Assume that (67) holds true. Then, for each and , we get where

Proof. The proof is similar to Lemma 4.
From Lemma 5, we see that
Hence, in view of the convolution formula (31) and the inversion of natural transform (33), inverting (85) yields where is the well-known Dirac delta function, and is given by (84). Then, using the basic property of the Dirac delta function, that is, , results in Integrating by parts gives where . Substituting from (87) gives with is given by (84). This result can be rewritten as where is the operator defined as

The integral in (94) is the Laplacian convolution formula for with . As a result, (93), together with (84) and (94), expresses the solution of Problem (34)–(37).

Remark 6. When and , for all , Problem (34)–(37) and its solution with defined as (88), (46), respectively, are reduced to that in Section 3 of [38].

2.1. Illustrative Examples

Here, we discuss two illustrative test cases to show the accuracy and effectiveness of our technique.

Example 1. Heat equation with zero temperatures at finite ends.
The following initial boundary value problem with homogeneous Dirichlet boundary conditions is a special case of the one-layer diffusion system (34)–(37) when and . For simplicity, we will take Thus, from (93), we have . From (46), we have , Hence, (54) yields So, we have at Moreover, as Therefore, (70) gives . Further, Finally, from (80), we get

which recovers the solution to problem (97) obtained via the separation of variables method in [55].

Example 2. Heat flow with sources and homogeneous boundary conditions.
The following initial boundary value problem is a special case of the one-layer diffusion system (34)–(37) when and . Thus, from (93), we have . From (46), we have , Hence, (54) yields So, has simple zeros at . Further, On the other hand, as . Therefore, (70) gives .
Finally, from (80), we get where with

When , formula (109) recovers the solution to problem (104) (when ) obtained via the eigenfunction expansion method in [55].

3. Multilayer Nonhomogeneous Diffusion System

Here, we are seeking the solution of our main problem defined in (2)–(8), which was converted into a sequence of initial boundary value problems (9)–(11). For the convenience 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 Section 3.2.

3.1. Solution of a Two-Layer Problem

For the two-layer problem, we have

Similar to what we denote in Section 2, we define the following vector notation , and

Also, analogues to (54), define

Further, similar to (67), suppose that there are nonzero simple roots and of the functions , respectively. That is,

Therefore, according to (84), we obtain

where can be defined as in Lemma 4.

Also, similar to (93), with the respective forms from Lemma 5 and the matching condition , we get where the operators are obtained from (94). The matching condition yields

For the unknown function , we can rewrite the linear integral equation (127) as where

Inspired by the convolution formula (31), the natural transform of (128) is which can be rewritten as

That is, where for which the inverse natural transform is where is the Dirac delta function. Hence, we have where for is the -times self-convolution of . Thus, one can conclude the solution to the bilayer diffusion problem (112)–(119) by the formulas (125) and (126), together with (123) and (124), with given in (135). Now, it is time to attack an illustrative example in the following subsection.

3.1.1. Illustrative Example

Here, we discuss the solvability of the following two-layer diffusion system.

Example 3. Temperature distribution in the two-layer slab with mixed boundary condition.
Consider the following initial boundary value system. The diffusion equations are the initial conditions are the outer boundary conditions are and the interface conditions at are Comparing this problem with the general one that is defined by (112)–(119) reveals Here, we will consider
Then, the solution of the problem (136)–(139) can be obtained from (125) and (126) as with

The operators are obtained from (94).

Next, we are going to simplify these formulas. Direct computations give

It is clear that when respectively. Moreover, as Therefore, (70) gives .

Further,

Thus,

Similarly, we have

So, (88) gives

Hence, loading the quantities in (141) and (142) gives the solution to problem (136)–(139) as

Applying the matching condition gives where

To solve this integral equation, we use (134), to obtain where is the Dirac delta function. Here, we used

Hence, from (135), we have

3.2. Solution of a Multilayer Problem

Here, we investigate the solvability of the main problem (2)–(8), through solving the initial boundary value problems (9)–(11). Similar to what we have denoted in Section 2, we consider the following notations: and, for all ,

Moreover, we define and let be the sequence of zeros of the function , i.e.,

Analogue to the computations of (84) and (93), we have for the current case, for all , where can be defined in a similar way as in Lemma 4, and with the respective forms defined by (87) in Lemma 5. This last equation (162) can be rewritten as in which and the linear operator is defined by for all .

The matching conditions , lead to

Using the matching conditions (12), we have for all . Thus, for ,

For ,

System (166), (167), and (168), of integral equations of the unknowns , can be adjusted as a matrix equation with is a tridiagonal matrix of order whose entries are as follows: and the vectors and are defined as

In fact, we can rewrite (169) as with and . In view of the convolution formula (31), the natural transform of (172) reads which is equivalent to where is the identity matrix. Once again, throughout the convolution sense (31), the natural transform inversion of (174) is where is the -times self-convolution of . Finally, the solution of the nonhomogeneous multilayer diffusion systems (9)–(11) and hence that of the main problem (2)–(8) is concluded as with the respective forms and defined as in (88) and (161), respectively, for all .

4. Conclusion

Throughout the current contribution, a one-dimensional -layer nonhomogeneous diffusion problem with time-varying data and general interface conditions has 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 applications. We have obtained the exact solutions for one- and multilayer nonhomogeneous diffusion problems. The former case has been solved by a new generalized integral transform; the later one (-layer problem) has been recast in a sequence of one-layer problems. The obtained results generalize and extend those in [31, 33, 38, 40, 42]. Our results motivate to deal with other types of diffusion problems, for example, reaction diffusion problems, advection-reaction diffusion problems, and nonautonomous reaction diffusion problems.

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 nonautonomous reaction diffusion PDEs. Those kinds of PDEs appear widely as epidemiological models to study and analyze the spread of diseases and pandemics [22–25].

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank the editors and the referees for their valuable comments and suggestions which improve the quality of the paper. This project was supported by the Academy of Scientific Research and Technology (ASRT), Egypt (Grant No. 6407).

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Copyright © 2022 Mohamed Akel et al. This 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.

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