19.1 Introduction

Generally, differential equations are the backbone of various physical systems occurring in a wide variety of disciplines viz. physics, chemistry, biology, economics, and engineering [1,2]. These physical systems are modeled either by ordinary or partial differential equations. Generally, in differential equations, the involved coefficients and variables are considered as deterministic or exact values. In that case, one may handle such differential equations (with deterministic coefficients or variables) by known analytic or numerical methods [3–5].

In actual practice, due to errors in experimental observation or due to truncation of the parametric values, etc. we may have only imprecise, insufficient, or incomplete information about the involved parameters of the differential equations. So, the parametric values involved in such differential equations are uncertain in nature. As such, there is a need of modeling different physical problems with uncertain parameters. In general, these uncertainties may be modeled through probabilistic, interval, or fuzzy approach [4, 5 ]. Probabilistic methods may not be able to deliver reliable results at the required condition without sufficient data. Therefore, in the recent years, interval analysis and fuzzy set theory have become powerful tools for uncertainty modeling.

In this chapter, we present different approaches to handle differential equations with interval uncertainty.

19.2 Interval Differential Equations

Moore [6] first introduced the concept of interval analysis and computations in 1966. Thereafter, this concept has been successfully applied by various researchers for uncertainty analysis. Several books [ 6 –8] have also been written by different authors representing the scope and various aspects of interval analysis. These books give an extensive review of interval computations which may help the readers to understand the basic concepts of interval analysis. Various mathematical techniques have been developed by different authors to handle the differential equations with interval analysis. As such, differential calculus is studied by Chalco‐Cano et al. [9] for interval‐valued functions using generalized Hukuhara differentiability [10], which is often referred as the most general concept of differentiability for interval‐valued functions. Also, the Hukuhara concept has been utilized by Stefanini and Bede [ 10 ,11] in a more generalized way for interval‐valued functions and interval differential equations (IDEs). Differential transformation method (DTM) is applied by Ghazanfaria and Ebrahimia [12]. Recently, Chakraverty et al. developed new techniques and approaches to handle differential equations with uncertain parameters [13–15].

The next section presents the preliminary concepts of interval arithmetic and parametric form of interval numbers.

19.3 Generalized Hukuhara Differentiability of IDEs

One of the known definitions of difference and derivatives for interval‐valued functions was given by Hukuhara [16]. Further, the new concepts named as generalized Hukuhara differences have been examined in Refs. [ 11 , 12 ].

Let us consider two intervals and . Then, the Hukuhara difference [11] is defined as

where and .

19.3.1 Modeling IDEs by Hukuhara Differentiability

Let us consider a linear interval‐valued differential equation 11, 12 ] as

(19.1)

where

19.3.1.1 Solving by Integral Form

After applying integration to Eq. (19.1) the equivalent integral equation 10, 11 ] is obtained as

(19.2)

where δ is a small positive quantity.

Using the Hukuhara difference

Now, according to Eq. (19.2), one may obtain

(19.3)

19.3.1.2 Solving by Differential Form

By using the differential form [ 12 , 13 ] to Eq. ( 19.1 ), we obtain

(19.4)

with respect to the initial conditions , .

We obtain two situations from Eq. (19.4) as

• Case (i): If , then the possible differential equations result in
  (19.5)
• with respect to the initial conditions, , .
• Case (ii): If , then the differential equations reduce to
  (19.6)
• with respect to the initial conditions, , .

19.4 Analytical Methods for IDEs

As mentioned in the introduction, several numerical and analytical methods [14,17] are available to solve IDEs. Generally, analytical solutions have a significant role in proper understanding of various science and engineering problems [18]. Accordingly, below we present one of the new approaches to handle IDEs [ 14 , 15 , 17 ].

19.4.1 General form of nth‐order IDEs

Let us consider nth‐order IDE in general form given in Refs. [ 14 , 17 ] as

(19.10)

subject to the initial conditions

where c_i's are real constants and 's are interval values for 0 ≤ i ≤ n − 1.

Here, is the solution to be determined.

Now, Eq. (19.10) may be represented in terms of lower and upper bounds as

19.11

subject to interval initial conditions

From Eq. (19.11), three possible cases arise with respect to the sign of the involved coefficients. The possible three cases are given as [ 17 , 18 ]:

• Case (i): All the coefficients c_n − 1, c_n − 2, …, c₁, c₀ are positive.
• Case (ii): Coefficients c_n − 1⋯c_n − m are positive and c_{n − m − 1}, c_{n − m − 2}, …, c₁, c₀ are negative.
• Case (iii): All coefficients c_n − 1, c_n − 2, …, c₁, c₀ are negative.

The analytical methods to handle IDE ( 19.10 ) in all three possible cases mentioned above are presented in the next section.

19.4.2 Method Based on Addition and Subtraction of Intervals

In this section, we present the solution procedure based on addition and subtraction of intervals [18] to handle all the three possible cases of considered IDE given by Eq. ( 19.10 ).

Case (i): When all the coefficients c_n − 1, c_n − 2, …, c₁, c₀ involved in Eq. ( 19.10 ) are positive. Then, one may obtain the following equations in terms of lower and upper bound as

(19.12)

and

(19.13)

By adding Eqs. (19.12) and (19.13), one may obtain

19.14

subject to the initial conditions

For simplification, Eq. (19.14) is represented as

(19.15)

with respect to the initial conditions

where , , , and .

Similarly, subtracting Eqs. ( 19.12 ) and ( 19.13 ), one may obtain

19.16

Further, Eq. (19.16) is represented as

(19.17)

with respect to the initial conditions

where , , and

It may be noted that both Eqs. (19.15) and (19.17) are in crisp forms. So, one may easily obtain solutions

Then, by applying standard methods, the lower‐bound solution is obtained by adding u(x) and v(x) whereas the upper‐bound solution is obtained by subtracting u(x) and v(x).

Case (ii): When some of the coefficients c_n − 1,…, c_n − m are positive and c_{n − m − 1}, c_{n − m − 2},…, c₁, c₀ are negative [ 14 , 18 ] in Eq. ( 19.10 ), one may obtain the lower‐ and upper‐bound equations as

(19.18)

and

(19.19)

subject to the initial conditions

Now, by using the same procedure as in case (i) one may also obtain the required lower‐ and upper‐bound solutions for this case.

Case (iii): When all the coefficients c_n − 1, c_n − 2,…, c₁, c₀ involved in Eq. ( 19.10 ) are negative. Then, one may obtain the following lower‐ and upper‐bound equations 18,19] as

(19.20)

and

(19.21)

subject to the initial conditions

Similarly, as in case (i), one may obtain the required lower‐ and upper‐bound solutions for case (iii). For more detailed understanding of various analytical methods to handle IDEs, one may refer Refs. [ 14 , 15 , 17 , 19 ,20].

Exercise

1. 1 By using Hukuhara differentiation, solve the following problems:
   1. 
   2. 
2. 2 Solve the following IDEs by using addition and subtraction of interval numbers
   1. with respect to , , and
   2. with respect to ,

References

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