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In this paper, we focus on the intermediate nodes of network and quantification of level of commodity and its cost on each node because intermediate nodes have stocking capacities which we generally see in the supply chain network. The commodity is supplied from a node to node in response to the power form of demand at a particular time. Since the traffic intensity of the demand of commodity also affects the flow of the commodity in the network, hence study of flow of commodity in the network is believed to be a significant contribution in this area. Several cases of quantifying the level of commodity in different situations as well as the cost analysis of incoming and outgoing commodity at a particular node have been thoroughly discussed in the paper. The present problem, presumably seeks to contribute to managerial decision making in supply chain network.

A fresh attempt needs to be made to quantify the level of commodity on the nodes of supply chain network at any instant of time with a particular amount of flow. Here, we seek to characterize the nodal properties of the network which in turn helps in making the strategies for future plans about a particular node of network and gauge the degree of importance of a particular node in whole network. Node wise quantification commodity paves the way for obtaining the economic inflow quantity at the node as well as the total optimal cost of the network system. In this sequence of knowledge, we wish to report some important developments which are relevant to it. Ahuja et al. [

1) Capacity constraints: x i j ≤ C ( i , j ) . The flow along an edge cannot exceed its capacity, ∀ i , j ∈ V .

2) Skew symmetric: x i j = − x j i . The flow from i to j is x i j and the flow from j to i is x i j .

3) Flow conservation: ∑ j ∈ V x i j = 0 unless i = source or j = sink.

The net flow to a node is zero except for the source that “produces” flow and the sink which “consumes” flow.

This paper focuses on the third condition at the intermediate nodes of the network where summation of all incoming flow and outgoing flow is zero, but in some case of the supply chain network this conservation law at instantaneous incoming and outgoing of flow does not hold good. In this situation, value of the flow is given as below at the intermediate node which appears like source or sink for short length of time that is given below, where x i is called the amount of flow. The amount of flow at i th node is treated as variable in case of the random network of the supply chain. During the random alteration in incoming flow and outgoing flow of commodity at the i th node, the amount of flow changes with time.

To manage the fluctuation in the amount of flow at intermediate nodes require storage capacity. This phenomenon provides me with the insight to measure the level of commodity which will be stored during this random change in the network. In case of the information flow, we will be able to know about the level of information at particular time according to that we may solve the routing problem of information flow with performance objectives.

Consider a node in computer communication network that receives the packet of information from various sources and must forward them toward their unlimited destination. Generally, the node has a fixed or at least a maximum rate at which it can transmit data. Since the arrival of packet to a node will be quite random, the node will usually have some buffering capacity, allowing the node to temporarily store packets which it can forward immediately.

Robert [

conduction equation i.e. ∂ ∂ t u ( x , t ) = k ∂ 2 ∂ x 2 u ( x , t ) continues to constitute the

conceptual foundation on which rests the analysis of many physical, biological, and social systems. This equation of heat flow may be treated as the partial differential equation of the commodity flow which has most important role in this paper. Sifaleras [

The present paper focuses on the alteration of level of node wise flowing commodity due to random change in the incoming and outgoing flow of commodity at particular time as novel and useful contribution to the existing knowledge. The demand rate of the commodity is in power form of available commodity with shape and size parameter. Here, we seek to measure and compute the level of commodity at any node at a particular time in terms of shape, size parameters and amount of flow of commodity. The cost analysis of incoming and outgoing node wise flowing commodities has been discussed under the different cases of size parameter.

There are following notations and its abbreviation for whole paper

x i j = Amount of outgoing commodity from i th node to j th , ∀ j ∈ V .

x j i = Amount of incoming commodity to i th node from j th , ∀ j ∈ V .

x i = ∑ j | ( i , j ) ∈ E x i j − ∑ j | ( i , j ) ∈ E x j i = Amount of flow at i th node.

I 0 , t = Level of commodity at time t with zero amount of flow.

I ε , t = Level of commodity at time t with amount of flow ( ∈ > Capacity ).

I 0 , t = Level of commodity at initial time with amount of flow x.

I x , t = Level of commodity, at time t and amount of flow at x.

T = Tension at any time on the node (to manage the level of commodity).

δ x = Small change in amount of flow.

α = Shape parameter which measures the shape of commodity packets i.e. α > 0.

β = Size parameter which measures the size of commodity packets i.e. β > 0.

θ = Rate of deterioration or evaporation or obsolescence of the commodity.

∂ θ ∂ t = ϕ = Acceleration of deterioration or evaporation or obsolescence of the commodity.

D x , t = Demand at time t and x amount of flow.

ρ = Traffic intensity of commodity at node i.e. ratio of rate of incoming commodity with rate of outgoing commodity.

ξ = Component of tension in horizontal direction i.e. parallel to the amount of flow axis.

T 1 = Tension at any point P at ( x , t ) co-ordinate.

T 2 = Tension at any point Q at ( x + δ x , t ) co-ordinate.

C h j i = Carrying cost of commodity which is coming from j th nodes to i th node.

C O i j = Ordering cost per unit of commodity which is outgoing from i node to j^{th} nodes.

C p j i = Purchasing cost per unit of commodity by node i th from node j th .

C h i = Holding cost per unit of commodity at i th node.

TVC = Total variance cost of the network system.

q i * = The optimal inflow quantity at the i th node.

The following assumptions have been thoroughly used in the paper.

1) Let the behavior of the intermediate nodes (supplier nodes) are appearing to be change according to the change of the amount of flow. Either it may play the role of source or sink.

2) The number of incoming arc flow and outgoing arc flow are randomly changing with respect to time at an intermediate node.

3) Let the initial amount of flow on a node is zero at any time t. Hence, it implies that level of commodity is zero on it.

4) The level of a commodity at a node increases with the increase of the amount of flow later on it starts decrease with an increase in the amount of flow of commodity.

5) When the amount of flow become more than capacity, the availability level of a commodity, becomes zero. It may be understood as when the demand of the commodity becomes more than the available commodity, in this situation amount of flow becomes more than the capacity i.e. node is now at the level of saturation.

6) The commodity is coming gradually from source and supply to the other nodes. We find that the finite level of commodity is maintained at the supplier node for fulfill the demand.

7) The elastic behavior of level of commodity with respect to the amount of flow. Due to this elasticity, the change in the amount of flow x x + δ x at time t, node has sophisticated tension to manage the level of the commodity that changes I x , t to I x + δ x , t . This tension can be realized on the basis of mainly two forces which are working on level of commodity.

8) The intermediate node manages the flow of commodities to fulfill the demand with its available stock of commodity at an instant of time. This creates a managerial tension which may be realized by two forces that are working on level of commodity. One of them is due to deterioration rate or the rate of evaporation (or obsolescence) that is working as the rate of retardation on level of

commodity i.e. ∂ D x , t ∂ t = α I x , t β and another is the cause of the rate of demand of commodity, i.e. ∂ D x , t ∂ t = α I x , t β (power form of level of commodity).

9) These forces create a tension on the level of commodity. Let T_{1} and T_{2} be tensions, making ψ and ϑ angles from the horizontal axis parallel to the amount of flow axis.

10) Actually, this tension is a hypothetical tension and it may be measured as

T = ρ δ x ( ∂ 2 D ∂ t 2 ± ∂ 2 θ ∂ t 2 )

11) When level of commodity increases, then T = ρ δ x ( ∂ 2 D ∂ t 2 − ∂ 2 θ ∂ t 2 ) and when level of commodity decreases then T = ρ δ x ( ∂ 2 D ∂ t 2 + ∂ 2 θ ∂ t 2 ) .

If we resolve T into two components, we find that the component along “amount of flow” axis direction as below

− T 1 cos ψ + T 2 cos ϑ = 0 ⇒ ξ = T 1 cos ψ = T 2 cos ϑ

And another component T 2 sin ϑ − T 1 sin ψ = Force for uplift the level of commodity.

The boundary conditions for the level of commodity at a supplier node is assumed as below

1) I 0 , t = 0 2) I ε , t = 0 , ε > C = Capacity 3) I x , 0 = 0 or f (x)4)

( ∂ I x , t ∂ t ) t = 0 = 0 or g (x)

We have obtained the following partial differential equation of the level of commodity at a particular node

∂ x α β ρ ξ I x , t β − 1 ∂ i x , t ∂ t ± ∂ x ρ ξ ∂ ϕ ∂ t = ( T 2 sin ϑ − T 1 sin ψ ) ξ

I x , t β − 1 ∂ I x , t ∂ t ± 1 α β ∂ ϕ ∂ t = ( ξ α β ρ ) [ ( ∂ I x , t ∂ x ) x + δ x − ( ∂ I x , t ∂ x ) x ] ∂ x (1)

Let ξ α β ρ = C 2 and assume that deterioration rate ϕ = ∂ θ ∂ t = Constant i.e. ∂ ϕ ∂ t = 0

∂ I x , t ∂ t = C 2 I x , t β − 1 ∂ 2 I x , t ∂ x 2 (2)

∂ I x , t ∂ t = C 2 ∂ 2 I x , t ∂ x 2 , Let the solution of the differential equation I x , t = X ( x ) ⋅ T (t)

and applying method of separation of variable to solve the partial differential equation, we get

1 X d 2 X d x 2 = 1 C 2 T d T d t = k

There are three possible conditions for k .

1) If k = 0 we get, I x , t = A x + B where A and B are constants. It implies that level of commodity is independent of time only linear function of amount of flow. This is suitable for the steady state flow but not for the transient flow model.

2) For k < 0 , let k = − p 2 , we get I x , t = ( a cos p x + b sin p x ) e − p 2 c 2 t , where a , b and p are constants. This implies that the level of commodity is periodic with amount of flow as we as converges at zero when t → ∞ . This model is generally suitable for measurement of amount of energy level at supplier node.

3) For k > 0 = − p 2 , we get I x , t = ( d e p x + f e − p x ) e p 2 c 2 t , where d and f are constants. It shows that when t → ∞ , the level of commodity also tend to infinite. It is not valid for physical model.

∂ I x , t ∂ t = C 2 I x , t β − 1 ∂ 2 I X , T ∂ x 2 , I x , t = { C 2 x } ( 2 1 − β ) { ( β − 1 ) ( 2 ξ ( 1 + β ) α β ρ ( 1 − β ) 2 ( C 2 ) 2 t ) } 1 β − 1

I x , t = ( 2 ξ ( 1 + β ) α β ρ ( 1 − β ) ) 1 β − 1 ( x 2 t ) 1 1 − β (3)

Since, I x , t = ( 2 ξ ( 1 + β ) α β ρ ( 1 − β ) ) 1 β − 1 ( x 2 t ) 1 1 − β , so demand rate can be expressed as

below.

∂ D x , t ∂ t = α { 2 ξ ( 1 + β ) α β ρ ( 1 − β ) } β β − 1 ( x 2 t ) β 1 − β

It implies that

∂ D x , t ∂ t = α { 2 ξ ( 1 + β ) α β ρ ( 1 − β ) } β β − 1 ( x 2 t ) β 1 − β x 2 β 1 − β ⋅ t 1 − 2 β 1 − β + C

Let the initial condition be D 0 , 0 = 0 then C = 0,

Hence, D x , t = α { 2 ξ ( 1 + β ) α β ρ ( 1 − β ) } β β − 1 ( 1 − β ) ( 1 − 2 β ) x 2 β 1 − β ⋅ t 1 − 2 β 1 − β (4)

The above expression shows the relation between demand and amount of flow.

The level of commodity, for particular amount of flow at sink as well as for same amount of flow at the source, may be same.

If β < 1 then I x , t = ( α β ρ ( 1 − β ) 2 ξ ( 1 + β ) ) 1 1 − β ( x 2 t ) 1 1 − β

Economic Inflow quantity (Optimal Inflow quantity)

Since total incoming amount of flow is q i = ∑ j | ( i , j ) ∈ E x j i , it takes t time in

supply to other nodes. Here, we consider the instantaneous fulfillment of commodity at a node

Hence, C h j i 2 × q i × t = C o i j × t × ∂ D x , t ∂ t q i

If β < 1 then I x , t = ( α β ρ ( 1 − β ) 2 ξ ( 1 + β ) ) 1 1 − β ( x 2 t ) 1 1 − β

q i * = q i = C O i j C h j i ∂ D x , t ∂ t = x β 1 − β α C O i j C h j i { ( 1 − β ) α β ρ 2 ( β + 1 ) ξ t } β 1 − β

q i * = x β 1 − β α C O i j C h j i { ( 1 − β ) α β ρ 2 ( β + 1 ) ξ t } β 1 − β (5)

Cost analysis at a particular node of a network

We here minimize TVC as

Minimize T V C = ∑ i ∈ V ∑ j | ( i , j ) ∈ E C o i j x i j + ∑ i C h i I x i , t + ∑ j | ( i , j ) ∈ E ∑ i ∈ V C p j i x j i

Subject to constraints

∑ j | ( i , j ) ∈ E x i j − ∑ j | ( i , j ) ∈ E x j i = x i

I x i , t = { ( 1 − β ) α β ρ 2 ( β + 1 ) ξ } 1 1 − β ( x 2 t ) 1 1 − β

∑ i ∈ V x i = 0 , x i j ≥ 0 andx j i ≥ 0 , 0 ≤ x i ≤ ε .

If β > 1 then I x , t = { − ( β − 1 ) α β ρ 2 ( β + 1 ) ξ } 1 β − 1 ( x 2 t ) 1 1 − β

I x , t = { ( β − 1 ) α β ρ 2 ( β + 1 ) ξ } 1 β − 1 ( 1 x ) 2 β − 1 e [ 2 log ( i t 1 2 ) β − 1 ] (6)

Under the above condition, from the expression (6) it is clear that we get complex values which are not possible in the practical situations. But in the definite range of size parameter β i.e. 1 < β < 1.5 we get real values of I_{x}_{,t} as

I x , t = { ( β − 1 ) α β ρ 2 ( β + 1 ) ξ } 1 β − 1 ( t x 2 ) 1 β − 1

q * = q i = C O i j C h j i ∂ D x , t ∂ t = 1 x β β − 1 α { ( β − 1 ) α β ρ 2 ( β + 1 ) ξ } β β − 1 C O i j C h j i

Let β = 1 + 1 2 n , where n = 1 , 2 , 3 ⋯ .

Consequently, we get the series solution

I x , t = ∑ n = 1 ∞ { α ( 2 n + 1 ) ρ 4 n ( 1 + 4 n ) ξ } 2 n ( t x 2 ) 2 n ,where n = 1 , 2 , 3 ⋯

Economic inflow quantity (optimal inflow quantity)

q * = q i = C O i j C h j i ∂ D x , t ∂ t = 1 x β β − 1 α { ( β − 1 ) α β ρ t 2 ( β + 1 ) ξ } β β − 1 C O i j C h j i

Cost analysis at a particular node of a network

In this section, we minimize the total variance cost as

Minimize

T V C = ∑ i ∑ j | ( i , j ) ∈ E C O i j x i j + ∑ i C h i I x i , t + ∑ j | ( i , j ) ∈ E ∑ i C p j i x j i

Subject to the constraints

∑ j | ( i , j ) ∈ E x i j − ∑ j | ( i , j ) ∈ E x j i = x i

I x , t = { ( β − 1 ) α β ρ 2 ( β + 1 ) ξ } 1 β − 1 ( t x 2 ) 1 β − 1

∑ i = 1 x i = 0 , x i j ≥ 0 and x j i ≥ 0 , 0 ≤ x i ≤ ε

The sensitivity analysis seeks to study, the variational approach of various parameters involved in the model which is used as an important instrument to judge the validity of the model for its future application. The following observations are worth mentioning to discuss the sensitivity analysis of the model. Matlab has been used for computing the values and drawing the graphs in this section. Tables and graphs are arranged under Appendix at last.

From

From the

The observations are drawn from

But the optimal inflow quantity decreases when 1) the value of stress parameter increases for fixed amount of flow; 2) the increment occurs in the size parameter as well as amount of flow.

From the

The

In the

From

From

From

From the

In the

Upon deeply analyzing the node wise level of commodity in the supply chain and total minimum variance cost of the network, we have been able to draw close observations under the sensitivity analysis of the model. By which we conclude that this model is very sensitive about the size parameter of the system which proves the rationale of the value of the flow at any node as an important performance measure of the system. On the basis of level of commodity, we have computed the economic inflow quantities (optimal inflow quantity), and obtained the total minimum variance cost of the network under the subject to the conditions with non-negative constraints. These results have a vast spectrum of application in various fields of network such as network of communication, supply chain network and commodity (gas, oil and energy etc.) flow network. In order to control the flow of commodity (overflow on buffer stock), this model is very useful. This model paves the path to measure the level of commodity during flow in the random network. To measure and quantify the level of commodity at any node under uncertain environment of flow of commodity and to find the effect of the price on the flow of commodity would be our future research.

Mishra, P.P., Mishra, S.S., Yadav, S.K., Singh, R.S. and Kumar, R. (2017) Quantification of Node Wise Commodity in Supply Chain and Its Cost Analysis. American Journal of Operations Research, 7, 64-82. http://dx.doi.org/10.4236/ajor.2017.71005

0.100 | 100 | 0.1 | 100 | 0.3704 |

0.100 | 200 | 0.1 | 100 | 0.0133 |

0.300 | 100 | 0.1 | 100 | 0.0319 |

0.300 | 100 | 0.1 | 400 | 0.0044 |

0.300 | 100 | 0.1 | 500 | 0.0032 |

0.1 | 100 | 100 | 100 | 0.0620 |

0.1 | 200 | 100 | 100 | 0.1338 |

0.1 | 300 | 100 | 100 | 0.2100 |

0.4 | 400 | 400 | 400 | 0.0008 |

0.5 | 500 | 500 | 500 | 0.0000 |

0.1 | 100 | 0.7 | 10 | 10 | 500 | 325.9247 |

0.2 | 100 | 0.7 | 15 | 20 | 500 | 248.4395 |

0.3 | 100 | 0.7 | 20 | 30 | 500 | 175.8259 |

0.8 | 100 | 0.7 | 25 | 40 | 500 | 0.1312 |

0.8 | 100 | 0.7 | 30 | 50 | 500 | 0.2489 |

0.8 | 100 | 0.7 | 35 | 60 | 500 | 0.4286 |

0.8 | 100 | 0.7 | 40 | 70 | 500 | 0.6875 |

0.8 | 100 | 0.7 | 20 | 20 | 700 | 0.0620 |

0.8 | 100 | 0.7 | 25 | 20 | 800 | 0.1435 |

0.8 | 100 | 0.7 | 30 | 20 | 900 | 0.2838 |

0.8 | 100 | 0.7 | 50 | 20 | 500 | 2.7703 |

0.8 | 100 | 0.7 | 50 | 20 | 600 | 2.5755 |

0.8 | 100 | 0.7 | 50 | 20 | 700 | 2.4215 |

0.8 | 100 | 0.7 | 50 | 20 | 800 | 2.2955 |

0.8 | 100 | 0.7 | 50 | 20 | 900 | 2.1899 |

0.8 | 300 | 0.7 | 50 | 20 | 700 | 0.7264 |

0.8 | 400 | 0.7 | 50 | 20 | 800 | 0.9182 |

0.8 | 500 | 0.7 | 50 | 20 | 900 | 1.0949 |

0.8 | 900 | 0.1 | 50 | 20 | 500 | 1.1448 |

0.8 | 900 | 0.2 | 50 | 20 | 600 | 1.4043 |

0.8 | 900 | 0.3 | 50 | 20 | 700 | 1.5529 |

1.0833 | 600 | 600 | 600 | 0.00 |

1.1000 | 500 | 500 | 500 | 0.00 |

1.1250 | 400 | 400 | 400 | 0.000 |

1.2500 | 200 | 200 | 200 | 0.000 |

1.500 | 100 | 100 | 100 | 0.2250 |

1.5 | 600 | 100 | 100 | 1.1736 (e−10) |

1.08 | 600 | 600 | 0.1000 | 600 | 0000 |

1.08 | 600 | 600 | 0.4000 | 600 | 0000 |

1.08 | 600 | 600 | 0.5000 | 600 | 0.006 |

1.08 | 600 | 600 | 0.7000 | 600 | 0.0055 |

1.08 | 600 | 600 | 0.9000 | 600 | 0.8750 |

1.500 | 100 | 100 | 0.1 | 600 | 0.2250 (e−7) |

1.500 | 600 | 100 | 0.1 | 600 | 1.1736 (e−10) |

1.5 | 100 | 600 | 0.7 | 600 | 0.1102 (e−5) |

1.5 | 150 | 600 | 0.7 | 600 | 0.0218 (e−5) |

1.5 | 200 | 600 | 0.7 | 600 | 0.0069 (e−5) |

1.5 | 250 | 600 | 0.7 | 600 | 0.0028 (e−5) |

1.1 | 900 | 0.1 | 50 | 20 | 500 | 1.1448 |

1.2 | 900 | 0.2 | 50 | 20 | 600 | 1.4043 |

1.3 | 900 | 0.3 | 50 | 20 | 700 | 1.5529 |

Case I:

Case II:

Some proofs are given below

Case II When β ≠ 1 and Let ∅ = ∂ θ ∂ t = Constant

∂ I x , t ∂ t = C 2 I x , t β − 1 ∂ 2 I x , t ∂ x 2 , I x , t = { C 2 x } ( 2 1 − β ) { ( β − 1 ) ( 2 ξ ( 1 + β ) α β ρ ( 1 − β ) 2 ( C 2 ) 2 t ) } 1 β − 1

I x , t = ( 2 ξ ( 1 + β ) α β ρ ( 1 − β ) ) 1 β − 1 ( x 2 t ) 1 1 − β

Since, I x , t = ( 2 ξ ( 1 + β ) α β ρ ( 1 − β ) ) 1 β − 1 ( x 2 t ) 1 1 − β , so demand rate can be expressed as

below.

∂ D x , t ∂ t = α { 2 ξ ( 1 + β ) α β ρ ( 1 − β ) } β β − 1 ( x 2 t ) β 1 − β

It implies that

D x , t = α { 2 ξ ( 1 + β ) α β ρ ( 1 − β ) } β β − 1 ( x 2 t ) β 1 − β x 2 β 1 − β ⋅ t 1 − 2 β 1 − β + C ⇒

Let the initial condition be D 0 , 0 = 0 then C = 0 ,

Hence, D x , t = α { 2 ξ ( 1 + β ) α β ρ ( 1 − β ) } β β − 1 ( 1 − β ) ( 1 − 2 β ) x 2 β 1 − β ⋅ t 1 − 2 β 1 − β

∂ I x , t ∂ t = C 2 I x , t β − 1 ∂ 2 I x , t ∂ x 2 , let I x , t = X ( x ) ⋅ T ( t ) be a solution and use the method of

separation of variables to solve the partial differential equation. We get

X β d T d t = C 2 T 2 − β d 2 X d x 2 and 1 C 2 T 2 − β d T d t = 1 X β d 2 X d x 2 = l which in turn gives

us

T = [ ( β − 1 ) ( C 2 l t + c 1 ) ] 1 β − 1

1 X β d 2 X d x 2 = l

If l = ( β + 1 ) ( 1 − β ) 2 ( C 2 ) 2 then X = { C 1 + C 2 x } ( 2 1 − β ) is a solution to differential

equation.

Further,

1 C 2 T 2 − β d T d t = l ⇒ T = [ ( β − 1 ) ( C 2 l t + C 3 ) ] 1 β − 1

I x , t = { C 1 + C 2 x } ( 2 1 − β ) [ ( β − 1 ) ( C 2 l t + C 3 ) ] 1 β − 1

I 0 , t = { C 1 } ( 2 1 − β ) [ ( β − 1 ) ( C 2 l t + C 3 ) ] 1 β − 1 = 0 ⇒ C 1 = 0

I x , 0 = { C 2 x } ( 2 1 − β ) [ ( β − 1 ) ( C 3 ) ] 1 β − 1 = 0 ⇒ C 3 = 0

I x , t = { C 2 x } ( 2 1 − β ) [ ( β − 1 ) 2 ξ ( β + 1 ) α β ρ ( β − 1 ) 2 ( C 2 ) 2 t ] 1 β − 1

I x , t = [ 2 ξ ( β + 1 ) α β ρ ( 1 − β ) ] 1 β − 1 ( x 2 t ) 1 1 − β

Since I x , t = [ 2 ξ ( β + 1 ) α β ρ ( 1 − β ) ] 1 β − 1 ( x 2 t ) 1 1 − β so demand rate can be expressed as

below.

∂ D x , t ∂ t = α { 2 ξ ( 1 + β ) α β ρ ( 1 − β ) } β β − 1 ( x 2 t ) β 1 − β

D x , t = α { 2 ξ ( 1 + β ) α β ρ ( 1 − β ) } β β − 1 ( x 2 t ) β 1 − β x 2 β 1 − β ⋅ t 1 − 2 β 1 − β + C

Let the initial condition be D 0 , 0 = 0 then C = 0,

D x , t = α { 2 ξ ( 1 + β ) α β ρ ( 1 − β ) } β β − 1 ( x 2 t ) β 1 − β x 2 β 1 − β ⋅ t 1 − 2 β 1 − β

When β ≠ 0 then there are following two cases arise as

Case-IIB ( If β > 1 ) , If β > 1 then

I x , t = { ( β − 1 ) α β ρ 2 ( β + 1 ) ξ } 1 β − 1 ( x 2 t ) 1 1 − β ( − 1 ) 1 β − 1 ⇒ I x , t = { ( β − 1 ) α β ρ 2 ( β + 1 ) ξ } 1 β − 1 ( 1 x ) 2 β − 1 e [ 2 log ( i t 2 ) β − 1 ]