The Decomposition Method for Linear, One-dimensional, Time-dependent Partial Differential Equations

The analytical solutions for linear, one-dimensional, time-dependent partial differential equations subject to initial or lateral boundary conditions are reviewed and obtained in the form of convergent Adomian decomposition power series with easily computable components. The efficiency and power of the technique are shown for wide classes of equations of mathematical physics.

In recent years, the Adomian decomposition method (ADM) has been applied to wide classes of stochastic and deterministic problems in many interesting mathematical and physical areas, [5,6].For linear PDEs, this method is similar to the method of successive approximations (Picard's iterations), whilst for nonlinear PDEs, is similar to the homotopy or imbedding method, [24].The ADM provides analytical, verifiable, and rapidly convergent approximations which yield insight into the character and behaviour of the solution just as in the closed-form solution.In this study, we review and develop new applications of the ADM for solving linear PDEs of the type (1.1) subject to the initial conditions (1.2), or to the lateral boundary conditions (1.3).
A wide range of linear PDEs, which have very important practical applications in mathematical physics, (see [35]), are investigated which include the advection equation (Section 4.1), the heat equation (Section 4.2), the wave equation (Section 4.3), the KdV equation (Section 4.4), and the Euler-Bernoulli equation (Section 4.5).Extensions to systems of linear PDEs and nonlinear PDEs, (see [20]) are presented in Sections 5 and 6, respectively.Finally, conclusions are presented in Section 7.

Adomian's decomposition method
First, let us define the following differential operators: with the convention that G 0 = F 0 = I = the identity operator.

A special case
We consider the special case of (1.1) with α n = 0 for n = 0,(N − 1), β m = 0 for m = 0,(M − 1), f = 0, α N , β M nonzero constants, given by Then (2.9) and (2.10) simplify to respectively.Solving (3.2), we obtain respectively.Then (2.11) gives explicitly the ADM partial t-solution of (1.2) and (3.1) as and the ADM partial x-solution of (1.3) and (3.1) as These solutions will be equal only when the compatibility conditions and the partial x-solution of (1.3) and (3.1) as D. Lesnic 5

Applications
Without loss of generality, we may assume that N ≥ M.

4.1.
The advection equation (N = M = 1).In this application, we consider the timedependent spread of contaminants in moving fluids, which, in the simplest case, is governed by the one-dimensional linear advection equation where β 1 is the constant coefficient of advection, which corresponds to the case then (3.4) gives the ADM partial t-solution whilst if (4.1) is solved subject to the boundary condition then (3.5) gives the ADM partial x-solution (see [8]) Example 4.1.Taking β 1 = 1, g 0 (x) = x, f 0 (t) = t, then both the ADM partial solutions (4.3) and (4.5) give, with only two terms u = u 0 + u 1 in the decomposition series (2.11), the exact solution u(x,t) = x + t of problem (4.1), (4.2), and (4.4).It is worth noting that this solution can also be obtained by using the ADM complete solution (see [1]) based on the recursive relationship using (2.11), that is, 4.1.1.The reaction-advection equation.We consider the linear reaction-advection equation where β 1 , α 0 are constants, which corresponds to the case ) is solved subject to the initial condition (4.2), then (2.9) gives Calculating a few terms in (4.9), we obtain and in general where 11) gives the ADM partial t-solution of problem (4.2) and (4.8) as If now (4.8) is solved subject to the boundary condition (4.4), similarly as above one obtains the ADM partial x-solution given by

The heat (diffusion) equation
. Consider the linear heat equation where β 2 > 0 is the constant coefficient of diffusion, which corresponds to the case If (4.14) is solved subject to the initial condition (4.2), then (3.4) gives the ADM partial t-solution of the characteristic Cauchy problem for the heat equation, namely, D. Lesnic 7 whilst if (4.14) is solved subject to the lateral boundary conditions then (3.5) gives the ADM partial x-solution of the noncharacteristic Cauchy problem for the heat equation (see [33]) (4.17) The solution (4.15) represents a simplified improvement over the Green formula and was previously obtained in [15] using the method of separating variables.

The reaction-diffusion equation.
We consider the biological interpretation of (4.14) with a linear source where β 2 > 0, α 0 are constants, which corresponds to the case 1).In contrast to the simple diffusion (α 0 = 0, see (4.14)), when reaction kinetics and diffusion are coupled through the term α 0 u, travelling waves of chemical concentration u may exist and can affect a biological change much faster than the straight diffusional process, see [34].If (4.18) is solved subject to the initial condition (4.2) then, similarly as in Section 4.1.1,one obtains the ADM partial t-solution given by On the other hand if (4.18) is solved subject to the boundary conditions (4.16), then (2.10) gives Calculating a few terms in (4.20), we obtain x 5  5! , (4.21) and in general Then (2.11) gives the ADM partial x-solution of problem (4.2) and (4.18) as given by For particular cases of f 0 , f 1 , and g 0 , one can calculate the series (4.19) and (4.23) explicitly, see [36].

The advection-diffusion equation. Taking
which arises in advective-diffusive flows when analysing the mechanics governing the release of hormones from secretory cells in response to a stimulus in a medium, flowing past the cells and through a diffusion column, see [38].In (4.24), β 2 > 0 is the diffusion coefficient, u is the concentration of hormones, and −β 1 > 0 is the flow velocity down the column.A similar situation arises in forced convection cooling of flat electronic substrates, (see [19]) or in the dispersion of pollutants in rivers.
For β 1 = constant, the ADM partial t-solution of problem (4.2) and (4.24) is given by (see [32]) where whilst the ADM partial x-solution of problem (4.16) and (4.24) is given by where and consider the initial and boundary conditions ) Then using (4.26) and (4.28), we obtain Using Leibniz's rule of product differentiation, we obtain ) Introducing (4.33) into (4.25),we obtain the ADM partial t-solution of problem (4.29) and (4.30) as Also introducing (4.34) into (4.27),we obtain the ADM partial x-solution of problem (4.29) and (4.31) as x, as required.Also, for obtaining the ADM partial t-solution, one can use directly the recursive relation (2.9) for problem (4.29) and (4.30) to obtain u 0 (x,t) = g 0 (x (2.11) gives the exact solution u = u 0 + u 1 = e x − x + t in only two terms.From this, it can be seen that directly applying the ADM to (4.29) produces a faster convergent series solution than (4.35) and (4.36).

The wave equation (N
. Consider the linear wave equation where β 2 > 0 is the square of the wave speed, which corresponds to the case then (3.4) gives the ADM partial t-solution, (see [42]) whilst if (4.37) is solved subject to the boundary conditions (4.16), then (3.5) gives the partial x-solution Particular examples of problem (4.37) and (4.38) solved using the ADM can be found in [14,17,45,48].Note that if we take β 2 = −1 in (4.37), we obtain the two-dimensional Laplace equation, which has been dealt with using the ADM elsewhere, see [12].

The telegraph equation. Consider the linear wave (telegraph) equation
which corresponds to the case D. Lesnic 11 If (4.41) is solved subject to the initial conditions (4.38), then (2.9) gives whilst if (4.41) is solved subject to the boundary conditions (4.16), then (2.10) gives and consider the initial and boundary conditions Calculating the initial term (4.42), we obtain Observing that the starting term (4.47) can be decomposed into two parts, namely, then a slightly modified recursive algorithm can be used instead of (4.42) (see [43]), namely, and so forth.Then (2.11) gives the ADM partial t-solution which can be verified through substitution to be the exact solution of (4.44) and (4.45).
The solution (4.50) was also previously obtained in [29] using the classical ADM based on (4.42) with the starting term (4.47), but the calculus in [29] is more complicated.Calculating now the initial term in (4.43), we obtain Similarly as before, by observing that the starting term (4.51) can be decomposed into two parts, namely, we use and thus Then the exact solution (4.50) of (4.44) and (4.46) is obtained with only three terms u = u 0 + u 1 + u 2 in the decomposition series (2.11).Note that if we take β 2 = −1 in (4.41), we obtain the two-dimensional steady-state diffusion equation with advection in the t-direction.

The linear Klein-Gordon equation. Consider the linear Klein-Gordon equation
which corresponds to the case 1) especially when the linear term α 0 u in (4.54) is replaced by a nonlinear function, the Klein-Gordon equation plays an important role in the study of solutions in condensed matter physics, (see [16]) and in quantum mechanics and relativistic physics; see [46].If (4.54) is solved subject to the initial conditions (4.38), then (2.9) gives and consider the initial and boundary conditions Applying (4.55), we obtain and in general (see [22]) Then (2.11) gives the ADM partial t-solution which can be verified through substitution to be the exact solution of (4.57) and (4.58).
Applying now (4.56), we obtain and in general we observe that and consider the initial and boundary conditions we use a slightly modified ADM instead of (4.55), namely, u 0 (x,t)=z 1 (x,t)=sin(x)sin(t), u 1 (x,t) = −t sin(x) + sin(x)sin(t) + G −1 2 (F 2 + 2I)u 0 (x,t) = 0, and in general with only one term.It can easily be verified that (4.71) is the exact solution of (4.66) and (4.67).The solution (4.71) was previously obtained in [21] using the classical ADM based on (4.55) with the starting term (4.69), but the calculus employed in [21] is more complicated.

The linear dissipative wave equation. Consider the linear dissipative wave equation
which corresponds to the case N = M = 2, α 0 = 0, α 2 = 1 in (1.1).If (4.74) is solved subject to the initial conditions (4.38), then (2.9) gives whilst if (4.74) is solved subject to the boundary conditions (4.16), then (2.10) gives and consider the initial and boundary conditions Calculating the first term in (4.75), we obtain which can be decomposed into two parts, namely, and use the modified ADM to give The decomposition method for linear PDEs and so on.We observe then that (2.11) gives the ADM partial t-solution u(x,t) = u 0 (x,t) + u 1 (x,t) + u 2 (x,t) It is easy to verify that (4.83) is the exact solution of (4.77) and (4.78).The solution (4.83) was previously obtained in [27] using the classical ADM based on (4.75) with the starting term (4.80), but the calculus in [27] is more complicated.
Calculating now the first term in (4.76), we obtain As before, splitting u 0 into two parts and replacing everywhere x with t and t with x in the above equation, we obtain that the ADM partial x-solution of problem (4.77) and (4.79) is equal to the exact solution (4.83).
Applying now (4.88), we obtain and so forth and in general we observe that based on (2.11), the ADM partial x-solution of problem (4.89) and (4.91) is given by as required; see also (4.93).
Example 4.8.Take β 1 = 0, β 3 = 1 and f (x,t) = 2e t−x in (4.85) to obtain the linear thirdorder dispersive, inhomogeneous equation and consider the initial and boundary conditions If (4.96) is solved subject to the initial condition (4.97), then (2.9) gives and so on.It is clear that the self-cancelling "noise" terms appear between various components, and keeping the noncancelled terms and using (2.11) lead immediately to the ADM partial t-solution (see [28]) which can be verified through substitution to be the exact solution of problem (4.96) and (4.97).
If now (4.96) is solved subject to the boundary conditions (4.98), then (2.10) gives D. Lesnic 19 and so on.In the above, one obtains self-cancelling "noise" terms appearing between various components of u 0 ,u 1 ,u 2 ,u 3 ,..., and keeping the noncancelled terms, and using (2.11) lead to the ADM partial x-solution of problem (4.96) and (4.98) as as required; see also (4.100).
It is worth noting that noise terms between components of the decomposition series will be cancelled, and the sum of these "noise" terms will vanish in the limit; see [10,41].
Alternatively, since the starting term u 0 in (4.101) can be decomposed into two parts, namely, then a slightly modified recursive algorithm can be used (see [43]), namely, and in general u k+1 (x,t) = F −1 3 G 1 u k (x,t) = 0 for all k ≥ 1.Then the exact solution (4.100) of (4.96) and (4.98) is obtained with only one term u = u 0 in the decomposition series (2.11).
Example 4.9.We consider the example tested in [26] obtained by taking At this stage, we note that the "exact" solution u(x,t) = e x + t obtained in [26] is incorrect since it does not satisfy (4.105).We remedy this mistake by taking the exact solution of (4.105) as which generates the initial and boundary conditions and so on.We observe that terms in the second term of u k cancel with the first term of u k+1 for k ≥ 1.Then (2.11) gives the exact solution (4.106), as required.At this stage, we note that the decomposition of u 0 in (4.110) is not unique, and for example, if one selects a better decomposition such as to obtain the exact solution (4.106) of problem (4.105) and (4.108) with only one term u = u 0 in the series (2.11).
Since the case N = 1, M = 4 of (1.1) is not a model of any well-known physical situation, it is not considered here, although one may think of it as a fourth-order diffusion process, see [25].
Since the case N = 1, M = 5 of (1.1) is not a model of any well-known physical situation, it is not considered here, although one may think of it as a linear fifth-order KdV equation, see [30].Finally, we mention that the case N = 1, M = 6 of (1.1) may be thought of as a model equation for linear seismic waves, see [7].The ADM described in this study can also be applied to these equations.

Extension to nonlinear PDEs
The ADM can also be extended to solving initial or boundary value problems for nonlinear, one-dimensional, time-dependent PDEs of the form where ∂ n t u = ∂ n u/∂t n for n = 0,(N − 1), and ∂ m x u = ∂ m u/∂x m for m = 1,(M − 1).Equation (6.1) has to be solved subject to the initial conditions (1.2), or to the lateral boundary conditions (1.3).Then, similarly as in (2.9) and (2.10), the ADM partial t-solution of problem (1.2) and (6.1), and the ADM partial x-solution of problem (1.3) and (6.1) will be given by the decomposition series (2.11), where the components of the series are D. Lesnic 25 calculated recursively from the following relationships: ) respectively.In (6.2) and (6.3),A k and B k are called the Adomian polynomials.These polynomials can be calculated for all forms of analytical nonlinearities, according to specific algorithms given, for example, in [6,37] as λ j G 0 u j ,..., , k ≥ 0. (6.5) For example, if A(u) = u 2 , then A k = k l=0 u l u l−k .
[33]0 in the the series(2.11).Some idea about appropriate choices in the decomposition of u 0 have been recently discussed by Lesnic and Elliott[33]who proposed a two-step ADM in which various parts of u 0 are tested if they satisfy the governing equation and/or the initial and/or boundary conditions.