Proper Generalized Decomposition Method for Solving Fisher-Type ...

ISSN 2070-0482, Mathematical Models and Computer Simulations, 2018, Vol. 10, No. 1, pp. 120–133. © Pleiades Publishing, Ltd., 2018.

Proper Generalized Decomposition Method for Solving Fisher-Type Equation and Heat Equation1 Chukwuemeke William Isaac Dept. of Mechanical Engineering, University of Ibadan, Ibadan, Nigeria e-mail: [email protected] Received May 11, 2017

Abstract⎯A model reduction technique—the Proper Generalized Decomposition (PGD) for solving time dependent and multidimensional parameters is reviewed and applied to both the Fisher-type equations and the heat equation. Space-time discretization and separated representation technique for obtaining fast convergence computation while maintaining real time is detailed. Three situations of the Fisher-type equation are solved by the PGD and the results show a perfect agreement with the exact solutions. The source term of the heat equation is given a Huxley source and the thermal diffusivity is taken to be linearly dependent of the spatial parameter. The results show how the Fisher-type equation finds application to the heat equation and that the PGD method allows a perfect representation of the temperature distribution defined in a 5-D tensorial product space and time. Keywords: Proper generalized decomposition, real time, model reduction, heat equation, Fisher equation, separated representation, space-time discretization DOI: 10.1134/S2070048218010039

1. INTRODUCTION The nonlinear nature of many physical problems encountered during modelling of both ordinary and complex engineering systems have attracted attention in recent years. Numerical methods, analysis and techniques are increasingly becoming popular to circumvent the complexity in constitutive behaviour and coupling of physical systems. An a priori numerical technique called the Proper Generalized Decomposition (PGD) was first developed by Chinesta et al. [1] to address the problem of the so called “curse of dimensionality” for problems defined in high dimensional spaces. It was shown by Ammar et al. [2] that this method can address a problem up to a configuration space of twenty dimensions. Also, in [3, 4] the authors illustrated how this method can alleviate the complexity arising from high dimensional space by it growing linearly with the dimension of the space as compared to the classical mesh based technique where the complexity grows exponentially with the dimension of the space and thus limiting numerical solutions up to configuration space of three dimensions. Another very important issue which the PGD method addresses is the difficulty in maintaining numerical stability for time dependent problems that has a wide range of characteristic times though not necessarily defined in high dimensional spaces [5]. Recent advances using the PGD method for solving multidimensional models [6] and in fluid dynamics for solving the Navier-Stokes equations [7, 8] have been performed. The Fisher equation first introduced by [9] is a time dependent problem used for gene propagation to determine the frequency of a mutant gene. Since then, different authors have modified this equation amongst is the Burger’s-Huxley equation [10] which describe the interaction between diffusion, convection and reaction. Modifications also include the introduction of new parameters so that it can find applications in many system processes for example in the diffusion of alleles in a population. Other application of the Fisher-type equation has been found in many physical problems such as in nuclear reactors, heat conduction, gas dynamics, elasticity, sound and waves. Different analytical and numerical approaches have been applied to solve the nonlinear Fisher-type equation [11–15]. Numerical methods such as the differential quadrature method [16], finite element method [17], finite difference method [18], iterative method [19] and method of line approach [20] have been used to find solutions to the Fisher’s type-equation. A major drawback is that these methods relied on low order discretization techniques of space and 1 The article is published in the original.

120

PROPER GENERALIZED DECOMPOSITION METHOD

121

f

tN

t

tN + 1

Fig. 1. Forward Euler numerical integration.

time. Stochastic techniques [21] and spectra method such as the collocation technique [22] are good tools that numerically solve linear or nonlinear partial differential equations of a given boundary value problem either in their weak or strong form. The spectra method approximates the solutions by high order orthogonal polynomial expansion thereby allowing a fewer degree of freedom. In spite of the various numerical techniques and the introduction of new parameters in the Fisher-type equation by the different authors, a major challenge is that as the parameters increase, numerical solutions become intractable because the complexity grows exponentially with the number of state space dimensions. In this paper, we introduce a discretization and high dimensional separated representation technique for the solution of the Fisher-type equation. The technique ensures a very fast convergence computation while maintaining real time. We derive the time-space equations and the equations involving all parameters introduced into the Fisher equation. The goal of the paper is first to describe the Proper Generalized Decomposition using the Fisher and Huxley equations with succinct explanation of both the space and time discretization techniques; second to obtain PGD numerical solutions for the Fisher-type equation and lastly to use the PGD solution of the Fisher equation to solve the heat equation for a typical situation where the thermal diffusivity is linearly dependent of the spatial parameter and the source term taken to be Huxley source. The paper is thus organized as follows: section two and three explains respectively the main discretization and separated representation techniques that describe the PGD method. Section four applied the PGD method to derive the temporal, spatial and parametric Fisher-type equation. Section five give some numerical experiments of the present method and compare the results with the exact solutions. In section six, a description of the heat equation using the PGD method is detailed. 2. DISCRETIZATION TECHNIQUES In this section the two types of discretization used in this study for both temporal and spatial variables are briefly discussed. It should be noted that other discretization methods could be used. 2.1. Time Discretization The partial differential equation (PDE) of the time discretization is performed using numerical method of the forward Euler explicit scheme as depicted in Fig. 1. Numerical integration is implemented according to the following relation

u N +1 − u N = 0, Δt

(1)

where u N is the function of the PGD approximation, N is the number of terms. The step time Δ t = t N +1 − t N . MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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CHUKWUEMEKE WILLIAM ISAAC

2.2. Space Discretization This involves finding the elemental mass and stiffness matrices and then assembling them. 2.2.1. Mass matrix. The elemental mass matrix is given by

∫

M e = ρ NT N d Ω,

(2)

Ω

where ρ is the mass density, N is the shape function over the domain Ω and the superscript T is its transpose. Equation (2) is well suited for the PGD method in this study because during the updates of the numerical simulation, the mass matrix does not vary with time and will not be recomputed. 2.2.2. Stiffness matrix. Two situations are presented here. First, when the stiffness matrix does not depend on the spatial variable, the elemental stiffness matrix in this case is written as

d Ni dx

∫

K ie, j =

x

Nx

dN j dx. dx

(3)

d Ni d N j dxS j , dx

(4)

∑S j =1

j

Which can also be expressed as Nx

∑∫ dx

K ie, j =

j =1 x

where Ni and N j are the shape functions of nodes i and j respectively, S j is the horizontal displacement at node j . In the second case, the elemental stiffness is influenced by the spatial parameter written as Nx

d Ni d N j x dxS j . dx

∑∫ dx

K i, j = e

j =1 x

3. SEPARATED REPRESENTATION TECHNIQUE This concept was first introduced in [23] which address the well known curse of dimensionality. The PGD method enforces a separable structure of a function which can be approximated by the summation of their separable terms given as N

u

N

(t, x, k, m) = ∑Ft i (t ) F xi ( x ) Fki ( k ) Ft i ( m),

(5)

i =1

=u

N −1

N

(t, x, k, m) + ∑Ft N (t ) F xN ( x ) FkN ( k ) FmN ( m) .

(6)

i =1

In a similar way, if k is the thermal diffusivity of a heat equation and f is the source term, the separated representation of these terms in PGD are respectively given as

k (t, x, k, m) =

Nk

∑V

j t

(t )V xj ( x )V kj ( k )V mj ( m) ,

(7)

(t )W xr ( x )W kr ( k )W mr ( m) .

(8)

j =1

f (t, x, k, m) =

Nf

∑W

r t

r =1

The computational complexity with this representation makes the dimension of the spaces linear giving it an advantage over the classical approaches where their complexities grow with the dimension of the spaces. In this paper, we seek for a solution of the heat function u where the constant k is linearly depenMATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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dent of the spatial parameter x and the source term f is a Huxley source which is a function of the parameter m and the function u of the PGD. 4. PROPER GENERALIZATION DECOMPOSITION METHOD In this section, the PGD method is used to solve the Fisher-type equation. Consider the parametric Fisher equation

∂ u − κ ∂ 2u = m u 2 1 − u ( ) 2 ∂t ∂x

(9)

with κ being the diffusion coefficient, m the reaction coefficient, u the function sought for, t the time parameter and x the spatial parameter. The usual x and t coordinates are defined in the interval Ω × ζ , two new coordinates κ and m are introduced which are defined in the interval R × D so that (t, x, κ, m ) ∈ Ω × ζ × ℜ × D. Given the test function u* and residual r ( x ), the weighting function can be written as

∫

u = u* r ( x)d Ω. Substituting Eq. (9) into the weighting function to give the weak form Ω

2 ⎛ ⎞ u* ⎜ ∂ u − κ ∂ u2 − m u 2 (1 − u ) ⎟ dtdxd κd m = 0. t ∂ ∂x ⎝ ⎠ Ω×ζ×ℜ×D

∫

(10)

A searching solution is

u (t, x, κ, m) =

N

∑ T (t ) X i

i

( x ) K i ( κ) M i ( m).

(11)

i =1

If the approximation in Eq. (11) is known at N iterations, then at N + 1 iterations, the equation reads

u

N +1

N

(t, x, κ, m) = ∑Ti (t ) X i ( x ) K i ( κ) M i ( m) + RN +1 (t ) S N +1 ( x )V N +1 ( κ)W N +1 ( m) .

(12)

i =1

The new notations RN +1 (t ) S N +1 ( x ) V N +1 ( κ ) W N +1 ( m) in Eq. (12) compute respectively the functional product TN +1 (t ) X N +1 ( x ) K N +1 ( κ ) M N +1 ( m). By the separated representation discussed in section 3, the test function is

u* (t, x, κ, m) = R* (t ) S ( x ) V ( κ ) W ( m) + R (t ) S * ( x ) V ( κ )W ( m) + R (t ) S ( x ) V * ( κ )W ( m) + R (t ) S ( x )V ( κ ) W * ( m) .

(13)

The explicit dependence of the function is dropped out in the notation since it is already indicated in the subscript. By doing a linearization, we take random function of S , V and W to find R . The constants and integers can be found for example in one case. Computing for u* = R* SVW and collecting terms, Eq. (10) reads 2 ⎛ ⎞ 2 R* SVW ⎜ ∂ R SVW + R κ d S2 VW − m ( RSVW ) (1 − RSVW ) ⎟ dtdxd κd m dx ⎝ ∂t ⎠ Ω×ζ×ℜ×D

∫

N

⎛ ∂R ⎞ d 2S 2 = − R* SVW ⎜ i S iV iW i + Ri κ 2i V iW i − m ( Ri S iV iW i ) (1 − Ri S iV iW i ) ⎟ dtdxd κd m. ∂t dx ⎠ i =1 ⎝ Ω×ζ×ℜ×D

∫

∑

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Integrating over ζ × ℜ × D one obtains the notations ∂R

∫ R ∂t dt

R1 =

∫S

S1 =

∫R

2

2

d S

∫S

S3 =

3

Ωt 4

S 5i

i

∫S

S6 = i

Ωx

Ωt

R7i =

∫ RTi dt

S 7i =

2

Ωt

R8i =

=

Ωx

∫ RTi dt

2

d Xi dx 2 dx

V6 = i

S 8i =

3

Ωt

3

d κ

V 7i =

2

4

d κ

V 8i =

3

Ωx

d m

∫ mW

W4 =

3

d m

4

d m

Ωm

∫ VK i d κ

(15)

∫ WM i d m

=

W 5i

Ωm

∫ V κK i d κ

∫ WM i d m

W6 = i

Ωm

∫ VK i d κ 2

∫ W m M i d m

W 7i =

2

Ωκ

∫ SX i dx

2

∫ mW

W3 =

Ωκ

∫ SX i dx

∫W

W2 =

Ωκ

Ωx

∫ RTi dt

d κ

Ωκ

V 5i

dm

Ωm

∫V

V4 =

dx

∫ SX i dx

=

Ωt

R6 =

4

2

Ωm

∫V

V3 =

dx

Ωx

∂T = ∫ R i dt ∂t

2

Ωκ

∫S

S4 =

dt

Ωt

R5i

3

∫W

W1 =

Ωκ

Ωx

∫R

R4 =

d κ

Ωm

∫ κV

V2 =

Ωx

∫ R dt

2

Ωκ

∫ S dx 2 dx

S2 =

dt

Ωt

R3 =

∫V

V1 =

dx

Ωx

Ωt

R2 =

2

Ωm

∫ VK i d κ 3

∫ W m M i d m

W 8i =

3

Ωκ

Ωm

The temporal equation becomes

S1V1W1 R* dR dt + S 2V 2W 2 R* Rdt − S 3V3W3 R * R 2dt + S 4V 4W 4 R * R 3dt dt

∫

∫

Ωt

∫

Ωt

∫

Ωt

Ωt

⎛ ⎞ ⎜ S 5iV 5iW 5i R * dTi + S 6iV 6iW 6i R* Ti − S 7iV 7iW 7i R* Ti 2 + S 8iV 8iW8i R * Ti 3 ⎟ dt. = − ⎜ ⎟ dt i =1 ⎝ Ωt Ωt Ωt Ωt ⎠ N

∑

∫

∫

∫

(16)

∫

The time Eq. (16) can be solved using the time discretization technique discussed in section 3. In a similar way, computing for u* = RS * VW , integrating over Ω × ℜ × D and collecting terms we obtain the spatial equation

2

R1V1W1 S * Sdx + R2V 2W 2 S * d S2 dx − R3V3W3 S * S dx + R4V 4W 4 S * S dx dx

∫

∫

ζx

ζx

∫

2

ζx

∫

3

ζx

⎛ ⎞ 2 d Xi 2 3 i i i i i i ⎟ = − ⎜ R5iV 5iW 5i S * X i + R6iV 6iW 6i S * − R V W S * X + R V W S * X 7 7 7 8 8 8 i i dx. 2 ⎜ ⎟ dx i =1 ⎝ ζx ζx ζx ζx ⎠ N

∑

∫

∫

∫

(17)

∫

The spatial Eq. (17) can be solved using the space discretization technique as explained in section 3. Again, computing for u* = RSV * W , integrating over Ω × ζ × D and collecting terms we obtain the diffusion coefficient equation MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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Table 1. Numerical solution for NUEX1 t

x

uPGD

uexact

Error

0.1

0.2

0.110365113

0.110365113

1.357 × 10–10

0.4

0.098151216

0.098151216

2.914 × 10–10

0.6

0.088331198

0.088331198

4.035 × 10–10

0.8

0.081344002

0.081344002

6.647 × 10–10

1

0.077438266

0.077438266

8.448 × 10–10

0.2

0.135139423

0.135139423

1.278 × 10–10

0.4

0.123926761

0.123926761

2.655 × 10–10

0.6

0.114699006

0.114699006

3.974 × 10–10

0.8

0.107373855

0.107373855

6.398 × 10–10

1

0.101638422

0.101638422

8.078 × 10–10

0.2

0.168065378

0.168065378

1.203 × 10–10

0.4

0.155601387

0.155601387

2.297 × 10–10

0.6

0.145957546

0.145957546

3.448 × 10–10

0.8

0.136285959

0.136285959

5.745 × 10–10

1

0.129410235

0.129410235

7.016 × 10–10

0.5

1

∫

∫

∫

∫

R1S1W1 V * Vd κ + R2S 2W 2 V * κVd κ − R3S 3W3 V * V d κ + R4S 4W 4 V * V d κ ℜκ

ℜκ

2

ℜκ

3

ℜκ

⎛ ⎞ = − ⎜ R5i S 5iW 5i V * K i + R6i S 6iW 6i V * κK i − R7i S 7iW 7i V * K i2 + R8i S 8iW 8i V * K i3 ⎟ d κ. ⎜ ⎟ i =1 ⎝ ℜκ ℜκ ℜκ ℜκ ⎠ N

∑

∫

∫

∫

(18)

∫

Lastly, computing for u* = RSVW *, integrating over Ω × ζ × ℜ and collecting terms we obtain the reaction coefficient equation

∫

R1S1V1 W * Wd m + R2S 2V 2 Dm

∫ W *Wd m − R S V ∫ W *m W 3 3 3

Dm

2

d m + R4S 4V 4

Dm

∫ W *mW

3

dm

Dm

⎛ ⎞ = − ⎜ R5i S 5iV 5i W * M i + R6i S 6iV 6i W * M i − R7i S 7iV 7i W * m M i2 + R8i S 8iV 8i W * m M i3 ⎟ d m. ⎜ ⎟ i =1 ⎝ Dm Dm Dm Dm ⎠ N

∑

∫

∫

∫

(19)

∫

Equations (18) and (19) can be solved algebraically since they do not involve any differential operator. 5. NUMERICAL EXPERIMENTS In this section three numerical experiments designated as NUEX1, NUEX2 and NUEX3 are performed and compared with the exact solutions to demonstrate the accuracy of the PGD method. The absolute errors are tabulated by the difference between the numerical and exact solutions as shown in Tables 1–3. NUEX1. Consider the Fisher equation

∂ u − ∂ 2u = u 2 1 − u . ( ) ∂t ∂x 2 MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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Table 2. Numerical solution for NUEX2 t

x

uPGD

uexact

Error

0.1

0.2

0.251269476

0.251269476

5.124 × 10–10

0.4

0.233333333

0.233333333

7.456 × 10–10

0.6

0.217526096

0.217526096

8.784 × 10–10

0.8

0.206182441

0.206182441

9.127 × 10–10

1

0.198720633

0.198720633

8.794 × 10–9

0.2

0.348182334

0.348182334

5.108 × 10–10

0.4

0.333944741

0.333944741

6.949 × 10–10

0.6

0.320953330

0.320953330

7.872 × 10–10

0.8

0.309226903

0.309226903

8.579 × 10–10

1

0.299150318

0.299150318

9.018 × 10–10

0.2

0.473244919

0.473244919

4.617 × 10–10

0.4

0.460613149

0.460613149

6.634 × 10–10

0.6

0.448148368

0.448148368

7.245 × 10–10

0.8

0.435909659

0.435909659

7.977 × 10–10

1

0.424099241

0.424099241

9.059 × 10–10

uPGD

uexact

Error

–1

0.682393537

0.682393537

4.578 × 10–9

–0.6

0.619240717

0.619240717

4.767 × 10–9

–0.2

0.551068651

0.551068651

4.945 × 10–9

0.2

0.484502803

0.484502803

5.677 × 10–9

0.6

0.426196252

0.426196252

5.614 × 10–9

1

0.392936469

0.392936469

6.176 × 10–9

–1

0.723902341

0.723902341

2.173 × 10–9

–0.6

0.669782554

0.669782554

2.754 × 10–9

–0.2

0.613315805

0.613315805

3.823 × 10–9

0.2

0.559472982

0.559472982

4.078 × 10–9

0.6

0.512760386

0.512760386

4.751 × 10–9

1

0.475721664

0.475721664

5.614 × 10–9

–1

0.770837021

0.770837021

9.017 × 10–10

–0.6

0.731510739

0.731510739

9.311 × 10–10

–0.2

0.687895779

0.687895779

9.667 × 10–10

0.2

0.642619718

0.642619718

9.153 × 10–10

0.6

0.598544272

0.598544272

8.847 × 10–10

1

0.556741316

0.556741316

8.542 × 10–10

0.5

1

Table 3. Numerical solution for NUEX3 t 0.1

0.5

1

x


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(a)

127

(b) 0.25

0.20

0.20 u

u

0.25 0.15

0.15

0.10

0.10

0.05 1.0

0.05 1.0

0.5 t

0 0

1.0 0.6 0.8 0.4 x 0.2

0.5 t

0 0

0.6 0.2 0.4 x

0.8 1.0

Fig. 2. NUEX1 results at c = 1, time step Δ t = 0.000005 and mesh size N = 30 for (a) PGD method and (b) exact solution.

Subject to the following initial and boundary conditions as given by [23]

(

) ( )) ( ))

u ( x,0) = 1 − 1 tanh c + 1 2 ( x ) , 2 2 4 u ( 0, t ) = 1 − 1 tanh c + 1 2 − 1 t , 2 2 4 2 u (1, t ) = 1 − 1 tanh c + 1 2 1 − 1 t . 2 2 4 2 The exact solution was given as

(

(

(

(

(21) (22) (23)

))

(24) u ( x, t ) = 1 − 1 tanh c + 1 2 x − 1 t , 2 2 4 2 where c is a constant. The surface plot for both the PGD method and the exact solution is shown in Fig. 2. NUEX2. Consider the Fisher equation

∂u − ∂ u = u 1 − u . ( ) ∂t ∂x 2 Subject to the following initial and boundary conditions as given by [24] 2

(

)

(25)

2

u ( x,0) = 1 − 1 + tanh x , 4 2 6

)

(

2

u ( 0, t ) = 1 1 + tanh 5t , 4 12

(

(26) (27)

)

u (1, t ) = 1 − 1 + tanh 1 ( 6 − 5t ) . 4 12 2

(28)

The exact solution was given as

(

))

(

u ( x, t ) = 1 − 1 + tanh 1 ( − 5t + 6x ) . 4 12 The surface plot for both the PGD method and the exact solution is shown in Fig. 3. NUEX3. Consider the non-linear Fisher-type equation 2

∂ u − ∂ 2u = u 1 − u u − γ . ( )( ) ∂t ∂x 2 Subject to the following initial and boundary conditions as given by [25]

u ( x,0) =

(

(30)

)

1 + γ γ −1 γ −1 − tanh ( x) , 2 2 2 2


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(a)

(b) 0.5 0.4 0.3 0.2 0.1 1.0

u

u

0.5 0.4 0.3 0.2 0.1 1.0 0.5 t

0.6 0.2 0.4 x

0 0

0.8 1.0

0.5 t

0 0

0.6 0.2 0.4 x

0.8 1.0

Fig. 3. NUEX2 results with time step Δ t = 0.000005 and mesh size N = 30 for (a) PGD method and (b) exact solution.

(a)

(b) 1.0

0.8

0.8

0.6

0.6

u

u

1.0

0.4

0.4

0.2 1.0 0.5 t

0 –1

–0.5

0 x

0.5

1.0

0.2 1.0 0.5 t

0 –1

–0.5

0 x

1.0

0.5

Fig. 4. NUEX3 results at γ = 0.01, time step Δ t = 0.000005 and mesh size N = 30 for (a) PGD method and (b) exact solution.

( ( )) 1 + γ γ −1 γ −1 γ +1 u (1, t ) = + tanh ( 1+ t . 2 2 2 2( 2 ) )

u ( − 1, t ) =

1+ γ γ −1 γ −1 γ +1 − tanh 1− t , 2 2 2 2 2

(32) (33)

The exact solution was given as

( (

))

1 + γ γ −1 γ −1 γ +1 (34) − tanh x− t , 2 2 2 2 2 where γ is a constant. The surface plot for both the PGD method and the exact solution is shown in Fig. 4. From the three numerical experiments, the tabulated results show a perfect agreement of the numerical PGD solutions with the exact solutions. The absolute errors decrease as the time increases. u ( x, t ) =

6. APPLICATION OF FISHER EQUATION TO THE HEAT EQUATION We introduce how the solution of Fisher-type equation could be applied to solve the heat equation. A situation where the thermal diffusivity is linearly dependent of the spatial parameter and the source term is a Huxley source. The equation is given as

∂ u − a + bx ∂ 2u = m u 2 1 − u , ( ) 2 ( ) ∂t ∂x

(35)

where m u 2 (1 − u ) is a Huxley source term and m the reactivity constant. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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Equation (35) is subject to the following initial and boundary conditions

u ( x,0) = u0 ( x ) , u ( 0, t ) = ul , u ( L, t ) = ur . It is seen that κ now becomes a + bx ; ul is the heat function at the left endpoint boundary and ur is the heat function at the right endpoint boundary. For simplicity, the coordinate defined in the interval ℜ for κ is replaced by two coordinates ψ × ϕ so that (t, x, a, b, m ) ∈ Ω × ζ × ψ × ϕ × D. Where the variability domains are Ω = [0;2], ζ = [0;1], ψ = [0;1.5], ϕ = [0;0.5] and D = [0;0.1] with a mesh of 100 nodes. The weak form of Eq. (35) reads

2 ⎛ ⎞ u* ⎜ ∂ u − ( a + bx ) ∂ u2 − m u 2 (1 − u ) ⎟ dtdxdadbd m = 0. ∂x ⎝ ∂t ⎠ Ω×ζ×ψ×ϕ×D

∫

(36)

Following the procedures highlighted in section 4, computing for u* = R* SYVW and integrating over ζ × ψ × ϕ × D one obtains the notation. ∂R

∫ R ∂t dt

R1 =

∫S

S1 =

Ωt

∫R

R2 =

2

dt

∫R

2

d S

dt

∫R

2

d S

∫ Sx dx 2 dx

S3 =

∫S

S4 =

dt

Ωt

∫R

R5 =

S 6i

i

∫ RTi dt

i

i

S 8i =

∫ Sx

dt

S9 = i

Ωt i R10 =

∫ RTi Ωt

4

dx

Y5 = Y6i

=

2

dx

2

dx

Y7 = i

dt

i S10 =

d Xi dx

2

dx

∫ SX i dx 2

∫V

∫Y

Y8i =

2

da

i

3

Y10i =

db

2

db

da

V4 =

∫V

da

V5 =

∫V

2

db

∫ YAi da

=

V7 = i

3

db

V 8i =

2

V9 = i

4

db

3

V10i =

Ωa

2

∫ mW

W5 =

d m 3

d m

4

d m

Ωm

∫ VBi db

W 6i

(37)

∫ WM i d m

=

Ωm

∫ VBi db

∫ WM i d m

W7 = i

Ωm

∫ VbBi db

∫ WM i d m

W 8i =

Ωm

∫ VBi db 2

W9 = i

Ωb

∫ YAi da

d m

Ωm

Ωb

∫ YAi da

2

∫ mW

W4 =

Ωb

∫ YAi da

d m

Ωm

Ωb

∫ YaAi da

∫W

W3 =

Ωb

V 6i

2

Ωm

Ωb 4

∫W

W2 =

Ωb 3

∫W

W1 =

Ωm

∫ bV

V3 =

Ωa

Y9 =

2

Ωb

Ωa

∫ SX i dx Ωx

∫V

V2 =

Ωa

2

Ωx 3

da

Ωa

d Xi

Ωx 2

2

Ωa

Ωx

∫ RTi dt ∫ RTi

∫S

S7 =

Ωt

R9 =

Y4 =

Ωx

Ωt

R8i =

dx

∫ SX i dx

=

Ωt

R7 =

3

∫V Ωb

Ωa

Ωx

∂T = ∫ R i dt ∂t

V1 =

Ωa

∫S

S5 =

dt

Ωt

R6i

∫Y

Y3 =

Ωx 4

da

Ωa

Ωx 3

2

∫ aY

Y2 =

Ωx

2

∫Y

Y1 =

Ωa

∫ S dx 2 dx

S2 =

Ωt

R4 =

dx

Ωx

Ωt

R3 =

2

∫ W m M i d m 2

Ωm

∫ VBi db 3

W10i =

Ωb

∫ W m M i d m 3

Ωm

Note the additional rows of three and eight in Eq. (37) arising from the spatial influence of the thermal diffusivity. The temporal equation now becomes MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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S1Y1V1W1 R* dR dt + S 2Y 2V 2W 2 R* Rdt + S 3Y3V3W3 R* Rdt − S 4Y4V 4W 4 R* R 2dt + S 5Y5V 5W 5 R* R 3dt dt

∫

∫

Ωt

∫

Ωt

∫

Ωt

⎛

N

∫

Ωt

Ωt

∑ ⎜⎜ S Y V W ∫ R* dt + S Y V W ∫ R*T

=−

i =1

i i i 6 6 6

dTi

i 6

⎝

i i i 7 7 7

i 7

Ωt

(38)

i

Ωt

⎞ + S 8iY8iV 8iW 8i R* Ti − S 9iY9iV 9iW 9i R* Ti 2 + S10i Y10i V10i W10i R* Ti 3 ⎟ dt. ⎟ Ωt Ωt Ωt ⎠

∫

∫

∫

Computing for u* = RS * YVW , integrating over Ω × ψ × ϕ × D and collecting terms we obtain the spatial equation

2

2

R1Y1V1W1 S * Sdx + R2Y 2V 2W 2 S * d S2 dx + R3Y3V3W3 S * x d S2 dx dx dx

∫

∫

ζx

∫

ζx

ζx

∫

∫

− R4Y 4V 4W 4 S * S 2dx + R5Y5V 5W 5 S * S 3dx ζx

ζx

(39)

⎛ d 2X i = − ⎜ R6iY6iV 6iW 6i S * X i + R7iY7iV 7iW 7i S * ⎜ dx 2 i =1 ⎝ ζx ζx N

∑

∫

+ R8iY8iV 8iW 8i S * x ζx

∫

∫

⎞ 2 d Xi 2 i i i i i i i i 3 ⎟ dx. R Y V W S X − + R Y V W S * X * 9 9 9 9 10 10 10 10 i i 2 ⎟ dx ζx ζx ⎠

∫

∫

Computing for u* = RSY * VW , integrating over Ω × ζ × ϕ × D and collecting terms we obtain the constant value of the heat diffusivity equation

∫

∫

∫

R1S1V1W1 Y * Yda + R2S 2V 2W 2 Y * aYda + R3S 3V3W3 Y * Yda ψa

ψa

∫

ψa

∫

+ R4S 4V 4W 4 V * V 2da + R5S 5V 5W 5 V * V 3da ψa

ψa

(40)

⎛ = − ⎜ R6i S 6iV 6iW 6i Y * Ai + R7i S 7iV 7iW 7i Y * aAi ⎜ i =1 ⎝ ψa ψa N

∑

∫

∫

⎞ + R8i S 8iV 8iW 8i Y * Ai + R9i S 9iV 9iW 9i Y * Ai2 + R10i S10i V10i W10i Y * Ai3 ⎟ da. ⎟ ψa ψa ψa ⎠

∫

∫

∫

Computing for u* = RSYV * W , integrating over Ω × ζ × ψ × D and collecting terms we obtain the spatial coefficient of the heat diffusivity equation MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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(b)

(a) 5 4 Temperature

Temperature

5 4 3 2 1

0 2.0

x a b m

3 2 1

1.5

1.0 0.5 t

0 0

0.2

0.4 x

1.0

0.6 0.8

0

0.5

1.0

1.5

Domain

Fig. 5. (a) Heat distribution along the space region over time (b) temperature field defined along different space domain at t = 0.15 s.

∫

∫

∫

R1S1Y1W1 V * Vdb + R2S 2Y 2W 2 V * Vdb + R3S 3Y3W3 V * bV db ϕb

ϕb

2

ϕb

∫

∫

− R4S 4Y 4W 4 V * V 2db + R5S 5Y5W 5 V * V 3db ϕb

ϕb

(41)

⎛ = − ⎜ R6i S 6iY6iW 6i V * Bi + R7i S 7iY7iW 7i V * Bi ⎜ i =1 ⎝ ϕb ϕb N

∑

∫

∫

⎞ + R8i S 8iY8iW 8i V * bBi − R9i S 9iY9iW 9i V * Bi2 + R10i S10i Y10i W10i V * Bi3 ⎟ db. ⎟ ϕb ϕb ϕb ⎠ Lastly, computing for u* = RSYVW * , integrating over Ω × ζ × ψ × ϕ and collecting terms we obtain the reaction coefficient equation

∫

∫

∫

R1S1Y1V1 W * Wd m + R2S 2Y 2V 2 Dm

∫

∫ W *Wd m

Dm

∫

∫

∫

+ R3S 3Y3V3 W * Wd m − R4S 4Y 4V 4 W * m W 2d m + R5S 5Y5V 5 W * m W 3d m Dm

Dm

Dm

(42)

⎛ = - ⎜ R6i S 6iY6iV 6i W * M i − R7i S 7iY7iV 7i W * M i ⎜ i =1 ⎝ Dm Dm N

∑

+ R8i S 8iY8iV 8i

∫

W * M i + R9i S 9iY9iV 9i

Dm

∫

∫

∫

W * m M i2 + R10i S10i Y10i V10i

Dm

⎞ W * m M i3 ⎟ d m. ⎟ Dm ⎠

∫

7. RESULT AND DISCUSSION Figure 5a shows the distribution of heat along the temporal and spatial coordinates. The solution obtained by the PGD method also allow a perfect representation of the temperature field defined in space of the heat diffusivity κ (a linear function of a and b ) and also in space of the reaction coefficient m as MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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depicted in Fig. 5b. Many nonlinear physical models require that the parameters characterizing them are continuously modified during modelling thereby resulting into many simulations which make the whole process time consuming, ineffective and high computational cost. With the PGD method, one needs not to perform many simulations to obtain satisfactory results even when there is a limited knowledge of the characteristics of the model. Many parameters can therefore be added in form of coordinates to give a full description of the heat distribution. In gene propagation application, diffusion-reaction processes and heat transfer, the dimensions increase as more parameters are needed to fully characterize the system. The complexity of the solution therefore increases as more parameters are included. The use of the classical mesh based approach increases exponentially the complexity thereby making it difficult to find an accurate solution to the problem. With the PGD method, the complexity increases linearly making it flexible to represent all parameters influencing the Fisher equation, wave equation, diffusion or heat governing equation. The spatial modes for all the parameters included as new coordinates, will therefore be the same but will be represented in their respective domains. 8. CONCLUSION In this paper, a high dimensional order called the Proper Generalized Decomposition has been reviewed to solve the Fisher-type equation which is typically encountered in gene propagation application. One of the advantages of this method is that it gives an efficient solution of the nonlinear problem while maintaining real time. The method is also empowered with great potentials to solve problems defined in high dimensional spaces. Even when all sources of variability is included in the Fisher-type equation; the PGD method could give a perfect representation of all variables as additional coordinates. For example, the result of this study shows that it can represent the frequency of a mutant gene in time and in space of the diffusion coefficient and also in space of the reaction coefficient. This application was illustrated using the heat equation where the thermal diffusivity is linearly dependent of the spatial variable and the source term is taken as Huxley source. The application of the PGD in nonlinear science with respect to computational genetics, heat diffusion-reaction, heat transfer, to mention but a few, has therefore great future prospects. REFERENCES 1. F. Chinesta, P. Ladeveze, and E. Cueto, “A short review on model order reduction based on proper generalized decomposition,” J. Fluid. Eng. 18, 395–404 (2011). 2. A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings, “A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory of modeling complex fluids,” J. Non-Newton. Fluid. 139, 153–176 (2006). 3. A. Ammar, F. Chinesta, E. Cueto, and M. Doblaré, “Proper Generalized Decomposition of time-multiscale models,” Int. J. Numer. Meth. Eng. 90, 569–596 (2012). 4. F. Chinesta, A. Ammar, A. Leygue, and R. Keunings, “An overview of the proper generalized decomposition with applications in computational rheology,” J. Non-Newton. Fluid. 166, 578–592 (2011). 5. P. Allier, L. Chamoin, and P. Ladeveze, “Proper Generalized Decomposition computational methods on a benchmark problem: introducing a new strategy based on Constitutive Relation Error minimization,” Adv. Model. Simulat. Eng. Sci. 2 (17), 1–25 (2015). 6. F. Chinesta, A. Ammar, and E. Cueto, “Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models,” Arch. Comput. Method. E 201–204, 327–350 (2010). 7. A. Dumon, C. Allery, and A. Ammar, “Proper generalized decomposition (PGD) for the resolution of NavierStokes equations,” J. Comput. Phys. 230, 1387–1407 (2011). 8. A. Dumon, C. Allery, and A. Ammar, “Proper generalized decomposition method for incompressible NavierStokes equations with a spectral discretization,” Appl. Math. Comput. 219, 8145–8162 (2013). 9. R. A. Fisher, “The wave of advance of advantageous genes,” Ann. Eugen. 7, 355–369 (1936). 10. O. Y. Yefimova and N. A. Kudryashov, “Exact solutions of the Burgers-Huxley equation,” J. Appl. Math. Mec. 68, 413–420 (2004). 11. A. Verma, R. Jiwari, and M. Koksal, “Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions,” Adv. Diff. Eq. 2014 (229), 1–13 (2014). 12. Y. S. Hamed, M. S. Mohamed, and E. R. El-Zahar, “Analytical approximate solution for nonlinear time-space fractional Fornberg-Whitham equation by fractional complex transform,” Commun. Numer. Anal. 2015, 115– 124 (2015). MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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13. V. Daftardar-Gejji and S. Bhalekar, “Solving multi-term linear and non-linear diffusion wave equations of fractional order by Adomian decomposition method,” Appl. Math. Comput. 202, 113–120 (2008). 14. M. Herzallah and K. Gepreel, “Approximate solution to the time-space fractional cubic nonlinear Schrödinger equation,” Appl. Math. Model. 36, 5678–5685 (2012). 15. A. H. Bhrawy and M. A. Alghamdi, “Approximate solutions of Fisher’s type equations with variable coefficients,” Abstr. Appl. Anal. 2013, 1–10 (2013). 16. R. Mittal and R. Jiwari, “Numerical study of Fisher’s equation by using differential quadrature method,” Int. J. Inform. Syst. Sci. 5, 143–160 (2009). 17. G. J. Fix and J. P. Roop, “Least square finite element solution of a fractional order two-point boundary value problem,” Comput. Math. Appl. 48, 1017–1033 (2004). 18. S. E. Alhazmi, “Numerical solution of Fisher’s equation using finite difference,” Bull. Math. Soc. Am. 12, 27– 34 (2015). 19. S. Behzadi, “Numerical solution for solving Burger’s-Fisher equation by using iterative methods,” Math. Comput. Appl. 16, 443–455 (2011). 20. J. R. Branco, J. A. Ferraira, and P. Olivaira, “Numerical methods for the generalized Fisher-KolmogorovPetrovskii-Piskunov equation,” Appl. Numer. Math. 57, 89–102 (2007). 21. H. A. Ghany and M. S. Mohamed, “White noise functional solutions for the wick-type stochastic fractional Kdv-Burgers-Kuramoto equations,” Chin. J. Phys. 50, 619–627 (2012). 22. M. Zarebnia and S. Jalili, “Application of spectral collocation method to a class of nonlinear PDEs,” Commun. Numer. Anal. 2013, 1–14 (2013). 23. G. Beylkin and M. J. Mohlenkamp, “Algorithms for numerical analysis in high dimensions,” SIAM J. Sci. Comput. 26, 2133–2159 (2005). 24. G. Hariharan, K. Kannan, and K. R. Sharma, “Haar wavelet method for solving Fisher’s equation,” Appl. Math. Comput. 211, 284–292 (2009). 25. A. H. Bhrawy and M. A. Alghamdi, “Approximate solutions of Fisher-type equations with variable coefficients,” Abstr. Appl. Anal. 2013, 1–10 (2013).


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