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Extension of the differential transformation method to nonlinear differential and integro-differential equations with proportional delays
Advances in Difference Equations volume 2013, Article number: 69 (2013)
In this paper, the differential transformation method is applied by providing new theorems to develop exact and approximate solutions of nonlinear differential and integro-differential equations with proportional delays represented by nonlinear multi-pantograph equations. Some examples are given to demonstrate the validity and applicability of the present method and a comparison is made with existing results.
In this paper, we consider the following nonlinear differential and integro-differential equations with proportional delays:
where ℱ, , K are given functions with appropriate domains of definition, , , , .
Functional differential and integro-differential equations with proportional delays are usually referred to as pantograph equations or generalized equations, and, as well, they are often used to model some problems with aftereffect in mechanics and the related scientific fields [1–13]. Many typical examples such as stress-strain states of materials, motion of rigid bodies and models of polymer crystallization can be found in Kolmanovskii and Myshkis’s monograph  and the references therein.
2 Differential transformation method
The method that is developed in this work depends on the differential transformation method (DTM) introduced by Zhou  in a study of electric circuits. This method constructs a semi-analytical numerical technique that uses Taylor series for the solution of differential equations in the form of polynomials. It is different from the high-order Taylor series method which requires symbolic computation of the necessary derivatives of the data functions.
There is no need for linearization or perturbations, large computational work and round-off errors are avoided. It has been used to solve effectively, easily and accurately a large class of linear and nonlinear problems with approximations [16–25].
The differential transformation of the k th derivative of function is defined as follows:
where is the original function and is the transformed function. Differential inverse transformation of is defined as follows:
In fact, inverse transformation (2) implies that the concept of differential transformation is derived from the Taylor series expansion. Although DTM is not able to evaluate the derivatives symbolically, relative derivatives can be calculated in an iterative way which is described by the transformed equations of the original function.
From definitions (1), (2) we can derive the following.
Theorem 1 Assume that , , and , , are the differential transformations of the functions , , and , , respectively, then:
The proof of Theorem 1 is given in .
3 Main results
In this section, we state the fundamental theorems of this paper.
Theorem 2 Assume that , and are the differential transformations of the functions , and , respectively, and , . Then:
Proof (i) From equation (1), we get
where , thus
and using (1) we have
Using the Leibnitz formula, we obtain
where , , and hence
From equation (1), we have
From the proof of formula (i), we get
Analogously, from to previous part of the proof, we get
where , ; therefore
Then from (1), we have
The proof is complete. □
Theorem 3 Assume that , and are the differential transformations of the functions , and , respectively, and , . Then:
Proof (I) It is obvious that
From here and equation (1), we get
Similarly as in previous part (I), we have
where , . Then
Using equation (1), we obtain
Put , then from the previous part we get(3)
where , and
where , . From (3) and (4), we obtain
but for we get . Then, using equation (1), we get
The proof is complete. □
4 Numerical examples
Example 1 As a practical example, we consider the following pantograph delay equation:
Using the differential transformation method, the differential transform version of equation (5) is
and the differential transform version of the initial condition has the form . From equation (6), we obtain the recurrence system of equations
From system (7), we have
Using the inverse transformation rule (2), we obtain the following series solution:
The closed form of the above series solution is , which is the exact solution of equation (5).
The same equation was solved by Evans and Raslan  using the Adomian decomposition method with complicated calculations of Adomian’s polynomials. Ghomanjani and Farahi  solved equation (5) using the Bezier control points method and obtained only approximation solution in the form
Example 2 Consider the following delay differential equation of the third order:
Using the differential transformation method, the differential transform version of equation (8), we get
and the differential transform version of the initial conditions , , gives
From (9) we have
Solving recurrence equations (10), we get
where . From here and using the inverse transformation rule (2), we obtain series solution in the form
which is the exact solution of equation (8). Evans and Raslan  solved equation (8) using the Adomian decomposition method and obtained a sequence of approximate solutions in the form of triple integrals with Adomian’s polynomials that required many symbolic calculations to obtain approximate solutions of (8). Saeed and Rahman  also solved equation (8) using the differential transformation method, but they transformed equation (8) into a system of three differential equations, which is a uselessly complicated approach to solving equation (8).
Example 3 Consider the following delay differential equation of the second order:
Applying the differential transformation method to equation (11), we get
and for initial conditions , we have . From (12), we obtain recurrence equations
Solving recurrence equations (13), we have
Thus , . Using the inverse transformation rule (2), we obtain the exact solution .
Example 4 Consider the nonlinear pantograph-type integro-differential equation of the first order
Solving recurrence equations (15), we get
From here and using the inverse transformation rule (2), we obtain a series solution in the form
The closed form of the above series solution is , which is the exact solution of equation (14).
Example 5 Consider the following nonhomogeneous first-order integro-differential equation with proportional delay:
and for initial conditions , , we have , . Following the same procedure as in the above mentioned examples, we get
Similarly, we obtain , . Using the inverse transformation rule (2), we get the exact solution . A homogeneous form of equation (16) subject to initial conditions was solved by Abazari and Kilicman . They obtained the closed form of a series solution in the form .
In the present paper, we have shown that the differential transformation method can be successfully used for solving nonlinear differential and integro-differential equations with proportional delays. New theorems are introduced with their proofs, and as application some examples are carried out. The main advantage of this method is that it can be applied directly to functional differential and integro-differential equations without requiring linearization, discretization or perturbation. Another important advantage is that this method is capable of greatly reducing the size of computational work and, moreover, the proposed method reduces the solution of a problem to the solution of a system of recurrence algebraic equations.
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The first author is supported by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. The second author is supported by the Grant P201/11/0768 of the Czech Grant Agency (Prague).
The authors declare that they have no competing interests.
The authors have made the same contribution. All authors read and approved the final manuscript.
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Šmarda, Z., Diblík, J. & Khan, Y. Extension of the differential transformation method to nonlinear differential and integro-differential equations with proportional delays. Adv Differ Equ 2013, 69 (2013). https://doi.org/10.1186/1687-1847-2013-69
- Series Solution
- Delay Differential Equation
- Previous Part
- Adomian Decomposition Method
- Differential Transformation Method