A Transformation Method for Delta Partial Difference Equations on Discrete Time Scale

Haider, Syed Sabyel; Rehman, Mujeeb Ur; Abdeljawad, Thabet

doi:https://doi.org/10.1155/2020/3902931

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Special Issue

Discrete Fractional-Order Systems with Applications in Engineering and Natural Sciences

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 3902931 | https://doi.org/10.1155/2020/3902931

A Transformation Method for Delta Partial Difference Equations on Discrete Time Scale

Syed Sabyel Haider,¹Mujeeb Ur Rehman,¹and Thabet Abdeljawad^2,3,4

Guest Editor: Baogui Xin

Received03 May 2020

Revised28 May 2020

Accepted18 Jun 2020

Published10 Jul 2020

Abstract

The aim of this study is to develop a transform method for discrete calculus. We define the double Laplace transforms in a discrete setting and discuss its existence and uniqueness with some of its important properties. The delta double Laplace transforms have been presented for integer and noninteger order partial differences. For illustration, the delta double Laplace transforms are applied to solve partial difference equation.

1. Introduction

The origin of calculus of finite differences is found from Brook Taylor (1717), rather it was Jacob Stirling, who found the theory (1730) and introduce the delta symbol for the difference, which is in common use nowadays. The development on calculus of finite differences in the beginning of the nineteenth century by Lacroix and remarkable work of George Boole, Narlund, and Steffensen appeared later in the nineteenth century. Jordan discussed calculus of finite differences with the classical approach in [1]. In modern era, the focus of mathematician is to correlate the continuous and the discrete, to shape in comprehensive unified mathematics, and to eliminate ambiguity. The calculus of finite differences is applicable to both continuous and discrete functions. For difference equations, Bohner and Peterson treat the dynamic equations on time scales in [2] and get surprisingly different results from continuous counterpart. Some results can be found in [3–19] which has helped to construct the theory of discrete fractional calculus.

Coon and Bernstein [20–22] defined the double Laplace transforms (continuous) and investigated many properties. Debnath [23] modified the properties and use the double Laplace transforms (continuous) to solve functional, integral, and partial differential equations. Dhunde and Waghmare [24] discussed convergence and absolute convergence of the double Laplace transforms (continuous) and, by application of double Laplace transforms, presented the solution of Volterra Integropartial differential equation. For applications of triple, quadruple, and -dimensional Laplace transforms (continuous), we refer the readers to [25–27]. Goodrich and Peterson [10] developed discrete delta Laplace transforms analogous to Laplace transforms discussed by Bohner and Peterson [2] in the continuous case, to solve difference and summation equations with initial data by applying the delta Laplace transforms. The delta Laplace transforms is given for newly defined Hilfer difference operator [28] and substantial difference operator in [29]. Bohner et al. [30] generalized properties of the Laplace transforms to the delta Laplace transforms on arbitrary time scales and discussed translation theorems and transforms of periodic functions. Compatible discrete time Laplace transforms with Laplace transforms was introduced in [31]. Savoye [32] highlighted the importance of discrete time problems and relationship of transforms to Laplace transforms on time scale. Fractional double Laplace transform was introduced in [33]; during derivation of Corollary 1, authors neglected the violation of semigroup property of Mittage-Leffler functions, and a counter example for semigroup property of Mittage-Leffler functions is given in [34]. The qualitative analysis of delay partial difference equations is considered as discrete analog of delay partial differential equations by Zhang and Zhou [35]. For solving partial difference equations Ozpinar and Belgacem introduced discrete homotopy perturbation Sumudu transform method in [36]. For solving partial differential equations, double Laplace transform was applied in [37, 38].

Here, we introduce the delta double Laplace transforms similar to the one presented by Bernstein [20] in such a way that properties and expressions bear a resemblance to that appearing in Debnath [23] for the continuous calculus. The double convolution product, we consider in this article, resemble with the convolution product defined for delta calculus in [2, 10], but it differs from the one defined by Atici in [8]. We consider the problem with constant coefficients in two independent variables and solve by applying the delta double Laplace transforms to partial difference equations with initial data.

This paper is divided into five sections. In Section 2, we shall present basic definitions and results from discrete calculus. Definition, existence, uniqueness, and series representation of the delta double Laplace transforms are given in Section 3. Some properties of the delta double Laplace transforms are proved in Section 4. In Section 5, we present the delta double Laplace transforms of partial differences.

2. Preliminaries

For convenience, this section comprises of some basic definitions and results from discrete calculus for later use in the following sections. The functions we consider usually are defined on the set and , for fixed .

The following concepts are discussed in [10, 16].

Falling function is defined for positive integer by

The forward jump operator is defined for by .

The set of regressive functions is defined for by .

The circle plus sum of is given by .

The additive inverse of is given by for .

Definition 1. (see [10]). Assume and . Then, the delta exponential function is given byBy the empty product convention, for any function .

Example 1. If is a constant such that (that is ), then the delta exponential function for a constant is given byFor a particular choice of , that is, the initial point of the domain of definition,

Definition 2. (see [10]). Assume and are in ; then, the delta definite integral is defined byNote that the value of integral , depending on the set . Also, by the empty sum convention,The delta indefinite integral is defined by

Definition 3. (see [10]). Assume . Then, the delta Laplace transform of based at is defined byfor all complex numbers such that this improper integral converges.
Note that throughout this article, we take the delta Laplace transform at the initial point of the set , unless stated otherwise.
The following concepts are also discussed in [10, 16].

Definition 4. (see [10]). A function is of exponential order if there exist a constant and the following inequality:If is of exponential order, then converges absolutely for , which ensures the existence of the Laplace transform. Even though the converse in not true, we restrict ourselves to only exponential order functions. For , the following are useful expressions for the delta Laplace transform of based at :for all complex numbers such that this infinite series converges.

Example 2. If , then for , we have

Definition 5. (see [10]). Assume . The convolution product is defined byNote that by the empty sum convention .

Lemma 1. (convolution theorem, see [10]). Assume . If both and exist, then the delta Laplace transform of the convolution product is given by

Lemma 2. (see [10]). Assume two functions . Let such that , and we have the summation by parts formula:

Definition 6. The generalized falling function is defined in term of gamma function bygiven that the expression in the above equation is justifiable. It is convenient to take , whenever is natural number and is a zero or negative integer.

Definition 7. The discrete Taylor monomial based at is defined byand the order Taylor monomial is defined by

Lemma 3. (see [10]). The following hold for delta Laplace of Taylor monomial:(i), for , (ii), for , (iii), for , .

In the next definition, we consider only delta difference with increment 1, and do not bother the different operators that we will not be using here. One can find the details of Definition 8 in [1, 39].

Definition 8. Assume , a function of two independent variables. Then, the partial difference of with respect to , regarding as a constant is given byThe partial difference of with respect to , regarding as a constant, is given byPartial difference equation is an equation containing partial differences.
Note that . Followed by the rule for integer order difference operator , we adopt the symbol for partial differences as follows: and .

3. The Delta Double Laplace Transforms

In this section, we give abstract definition of the delta double Laplace transform. For convenience, we simplify definition to series representation followed by Goodrich and Peterson [10] simplification of the delta Laplace transform. Also, condition for existence, uniqueness, and linearity of the delta double Laplace transform has been revealed.

Definition 9. Assume . Then, the delta double Laplace transform of based at () is the successive application of the delta Laplace transform on and in any orderwhere and are the delta Laplace transforms (single) based at with respect to and , respectively, and is the delta double Laplace transform based at . The delta double Laplace transform of a function of two variables and is defined in - plane provided the following double sum converges:for all complex numbers and .
One can easily verify by using Lemma 4 that . Later, in Theorem 2, we will prove that the double infinite series is absolutely convergent. It is well known that absolutely convergent series behave nicely and change in the order of summation allowed. Therefore, we can operate in either way .

Lemma 4. Assume . Then,for all complex numbers and such that the infinite series converges.

Proof. By using the definition of the delta double Laplace transform, consider the following:Now, by the definition of delta integral from discrete calculus, we obtainIn preceding steps, we use the definition of delta exponential function and the fact that and , since and are regressive functions. In the following step, we use and to reindex the sums as follows:

Theorem 1. Assume functions , , and such that the delta double Laplace transforms exist, then the following holds:(i)(ii)(iii)

Proof. Under the assumption stated above and by Lemma 4,(i)For and , we have(ii)The proof is similar to part (i).(iii)For and , we have

Example 3. (i)If for , then ,(ii)If for , then .(iii)By Lemma 4(iv)By using Theorem 1 part (iii), we obtainBy using Lemma 3,If we choose either or , then as a special case of the aboveCoon and Bernstein [20, 21] defined the double Laplace transforms and discussed convergence and existence for the continuous case. We discuss discrete analogue of the double Laplace transforms.

Definition 10. A function is of exponential order with respect to and , respectively, if there exist a constant and such that, for each and , the inequality holds, where for and .

Theorem 2. If a function is of exponential order , then the delta double Laplace transform converges absolutely for and provided , .

Proof. Assume is of exponential order . Then, there exists a constant and such that, for each and , . Thus, for and , we consider the following:Since and , therefore and . Hence, the delta double Laplace transform of converges absolutely.
Theorem 2 ensures the existence of the delta double Laplace transform. In general, the converse does not hold. We should consider functions of some exponential order , to ensure the delta double Laplace transform of which does converge somewhere in the complex plane outside the both closed balls of radius , centered at , that is, we can choose for and .

Theorem 3. Suppose . If the delta double Laplace transform of converges for and , where , and let , , then the delta double Laplace transform of converges for , , and , converges for and .

Proof. Since and the delta double Laplace transform of converges for and , where . We have that, for and ,Theorem 3 exposed the linearity property of the delta double Laplace transform, and Theorem 4 revealed the uniqueness.

Theorem 4. Suppose and , . If , provided , , and , then for all .

Proof. By hypothesis, we havefor and . This implies thatfor and . Since, by Theorem 2, the double infinite series is absolute convergent, therefore comparison of both sides implies thatFor each fix and for all , this implies thatFor each fix , we obtain

4. Basic Properties of the Delta Double Laplace Transform

In this section, following Bohner et al. [30], we prove some properties of the delta double Laplace transform. We also define double convolution product of discrete functions followed by Goodrich and Peterson [10] convolution product (single) of discrete functions. We present, the delta double Laplace transform of double convolution product for later use to solve difference equations.

Theorem 5. Assume and exists. If , thenwhere is the Heaviside unit step function defined by

Proof. We have, by Lemma 4,Reindex by and ,In the last step, we use Lemma 4 with the fact and .
Theorem 5 gives different results from its continuous counterpart stated in [23]. We state the useful shifting Theorem 6 for discrete setting.

Theorem 6. Assume and exist. If , then(i).(ii), where is the Heaviside unit step function defined by

Proof. (i)We have by Lemma 4, Reindex by and ,(ii)By use of Lemma 4 and reindex by and ,

Theorem 7. Assume is periodic with and exist; then,

Proof. Under the assumption, we have, by Lemma 4,In the last step, we used and to reindex second double summation. In second double summation, periodicity of implies that

Definition 11. Assume . The double convolution product is defined byNote, by empty sum convention, .

Lemma 5. Assume . The double convolution product is commutative:

Proof. By Definition 11 and the change of variables and , we have

Theorem 8. (convolution theorem). Assume . If both and exist, then the delta double Laplace transform of double convolution product is

Proof. Under given assumption, we have, by Lemma 4 and the fact ,In the last step, we used Definition 11; next, making the change of variables and gives us thatIn the previous steps, we interchanged the order of first pairs and second pairs of summation and change variables and .

Corollary 1. Assume . If and and the delta Laplace transform exists, thenwhere the product on right- and left-hand sides is given by Definitions 5 and 11, respectively.

Proof. By double convolution theorem, we haveSince ,The last step is followed from single convolution Lemma 1.

5. The Delta Double Laplace Transforms of Partial Differences

In this section, we examine the action of the delta double Laplace transforms on first order partial differences. The results developed for first order partial differences are further used to establish properties of the delta double Laplace transforms of generalized order partial difference, similar to that appeared in [40] for fractional order partial derivatives. We usually consider functions , of exponential order with respect to and , respectively, ensuring that delta Laplace and the delta double Laplace transforms of and its partial differences does exist.

Lemma 6. Assume , such that the delta Laplace transforms exist for constants and . Then,

Proof. By definition of the delta Laplace transforms on ,Apply summation by parts (Lemma 2) on , and using the fact , we have thatUse the fact and ,Since ,Let . Consider the left-hand side of equation (61) and use the definition of delta difference:By using linearity property of the delta Laplace transforms, we obtainNow, consider the right-hand side of equation (61) and use :By using the definition of delta difference, we obtainEquality holds in equation (61) from equations (68) and (70). Proof of equations (60) and (62) is similar to proof of equations (59) and (61), respectively.

Theorem 9. Assume , such that the delta double Laplace transforms exist for constants and . Then,(i)(ii)

Proof. Since, by definition, the delta double Laplace transforms is the successive application of the delta Laplace transforms on and in any order, therefore .(i)Consider By using equation (59) of Lemma 6, we obtain Use linearity property of the delta Laplace transforms for ,(ii)The proof is similar to part (i).Note that, for constant , . We adopt the following symbols in our result which are nonzero, in general, and , that is, first we take difference and then evaluate at .

Lemma 7. Assume , such that the delta Laplace transforms exist for constants and . Then,(i)(ii)(iii)(iv)

Proof. (i)We prove this part by induction on , and result for has been proved in Lemma 6. Assume the result is true for : We will try to establish result for , beginning with the following: Let , and we have that Again using equation (59) of Lemma 6, By using assumption for , The result holds for , whenever it holds for . Hence, by induction, result in part (i) holds.(ii) Let , and use part (i) of the same Lemma:Proof of (ii) and (iv) is similar as proof of part (i) and (iii), respectively.

Theorem 10. Assume , such that the delta double Laplace transforms exist for constants and . Then,(i)(ii)(iii)

Proof. Since by definition, the delta double Laplace transforms is the successive application of the delta Laplace transforms on and in any order; therefore, .(i)Using Lemma 7 part (i) and linearity of Laplace, we consider the following:(ii)Proof is similar as in part (i).(iii)Using Lemma 7 part (iii) and linearity of Laplace, we consider the following:In the previous step, we used Lemma 7 part (iv). In the following step, using Theorem 10 part (ii),

Theorem 11. Assume . If , then for constants and , and we have

Proof. For , letThen, the difference isBy separating last term for , from the first double sum, we obtainBy separating last term for , from the first sum, we obtainNow, for constants and , taking the delta double Laplace transforms on both sides,By application of Theorem 10 (iii) for , . On right-hand side, we obtainIn the last step, are zero by empty sum convention, and on further simplification, we obtain

Example 4. (a)Solve the partial difference equation: Application of the delta Laplace transforms to initial conditions by Lemma 3, Apply the delta double Laplace transforms to difference equation and then use linearity property: Using Theorem 9, Inverting the delta Laplace transforms pairs(b)Solve the same partial difference equation with slightly different initial conditions: Application of the delta Laplace transforms to initial conditions by Lemma 3: Inverting delta Laplace transforms pairs,Assume , then the Riemann–Liouville fractional difference of order , for is given by , for . By using the discussion and results, from Theorem 2.65 to Theorem 2.70, in [10], we take the starting point of the double Laplace and , respectively, for sum and difference operator.

Corollary 2. Assume , such that the delta double Laplace transforms exists for constants and and denote . Then, for and , the delta double Laplace transforms of fractional order operators is given by

Proof. (i)Proof is an implication of Definition 3.1 and Theorem 2.67 of [10](ii)Result is obtained by application of Theorem 5.4 part (i) and Theorem 2.70 of [10](iii)Result is obtained by application of Theorem 5.4 part (ii) and Theorem 2.70 of [10]

Example 5. Solve the fractional difference equation for :Apply the delta Laplace transforms to initial condition . For , we have which implies and therefore , also . Application of the delta double Laplace transforms on both sides of fractional difference equation (103) and making use of equation (101) on left-hand side, and on the right-hand side we used Example 3 to obtainUsing and simplifying the above,Inverting the delta Laplace transforms pairs by making use of Theorem 1 (iii), together with Lemma 3 (iii),

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All authors contributed in writing review and editing the article and have read and agreed to the published version of the manuscript.

Acknowledgments

The authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), Group no. RG-DES-2017-01-17.

References

C. Jordan, Calculus of Finite Differences, AMS Chelsea Publishing, Chelsea, NY, USA, 1979.
M. Bohner and A. C. Peterson, Dynamic Equations on Time Scales an Introduction with Applications, Birkhauser Boston, Cambridge, MA, USA, 2001.
T. Abdeljawad and D. Baleanu, “On fractional derivatives with exponential kernel and their discrete versions,” Reports on Mathematical Physics, vol. 80, no. 1, pp. 11–27, 2017.
View at: Publisher Site | Google Scholar
T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011.
View at: Publisher Site | Google Scholar
T. Abdeljawad, D. Baleanu, F. Jarad, and R. P. Agarwal, “Fractional sums and differences with binomial coefficients,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 104173, 6 pages, 2013.
View at: Publisher Site | Google Scholar
T. Abdeljawad, S. Banerjee, and G. C. Wu, “Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption,” Optik, Article ID 163698, 2019.
View at: Publisher Site | Google Scholar
T. Abdeljawad, “On delta and nabla Caputo fractional differences and dual identities,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 406910, 12 pages, 2013.
View at: Publisher Site | Google Scholar
F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Differential Equations, vol. 2, no. 2, 2007.
View at: Google Scholar
A. Ivan, J. Losada, and J. J. Nieto, “On quasi-periodic properties of fractional sums and fractional differences of periodic functions,” Applied Mathematics and Computation, vol. 273, pp. 190–200, 2016.
View at: Publisher Site | Google Scholar
C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, New York, NY, USA, 2015.
C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 191–202, 2011.
View at: Publisher Site | Google Scholar
L.-L. Huang, J. H. Park, G.-C. Wu, and Z.-W. Mo, “Variable-order fractional discrete-time recurrent neural networks,” Journal of Computational and Applied Mathematics, vol. 370, p. 112633, 2020.
View at: Publisher Site | Google Scholar
G.-C. Wu, D. Baleanu, and W.-H. Luo, “Lyapunov functions for Riemann-Liouville-like fractional difference equations,” Applied Mathematics and Computation, vol. 314, pp. 228–236, 2017.
View at: Publisher Site | Google Scholar
G. C. Wu, Z. G. Deng, D. Baleanu, and D. Q. Zeng, “New variable-order fractional chaotic systems for fast image encryption,” Chaos, vol. 29, Article ID 083103, 2019.
View at: Publisher Site | Google Scholar
M. T. Holm, “The Laplace transform in discrete fractional calculus,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1591–1601, 2011.
View at: Publisher Site | Google Scholar
M. T. Holm, The Theory of Discrete Fractional Calculus: Development and Application, University of Nebraska-Lincoln, Lincoln, NE, USA, 2011.
S. S. Haider and M. U. Rehman, “Ulam-Hyers-Rassias stability and existence of solutions to nonlinear fractional difference equations with multipoint summation boundary condition,” Acta Mathematica Scientia, vol. 40, no. 2, pp. 589–602, 2020.
View at: Publisher Site | Google Scholar
S. S. Haider and M. U. Rehman, “Construction of fixed point operators for nonlinear difference equations of non integer order with impulses,” Fractional Calculus and Applied Analysis, vol. 23, no. 3, 2020.
View at: Google Scholar
F. Jarad, K. Taş, and K. Ta, “A new transform method in nabla discrete fractional calculus,” Advances in Difference Equations, vol. 2012, no. 1, 190 pages, 2012.
View at: Publisher Site | Google Scholar
D. L. Bernstein, “The double Laplace integral,” Duke Mathematical Journal, vol. 8, no. 3, pp. 460–496, 1941.
View at: Publisher Site | Google Scholar
G. A. Coon and D. L. Bernstein, Some Properties of the Double Laplace Transformation, Taylor Instrument Companies, University of Rochester Press, Rochester, NY, USA, 1953.
G. A. Coon and D. L. Bernstein, “Some general formulas for double Laplace transformations,” Proceedings of the American Mathematical Society, vol. 14, no. 1, p. 52, 1963.
View at: Publisher Site | Google Scholar
L. Debnath, “The double Laplace transforms and their properties with applications to functional, integral and partial differential equations,” International Journal of Applied and Computational Mathematics, vol. 2, no. 2, pp. 223–241, 2016.
View at: Publisher Site | Google Scholar
R. R. Dhunde and G. L. Waghmare, “On some convergence theorems of double Laplace transform,” Journal of Informatics and Mathematical Sciences, vol. 6, pp. 45–54, 2014.
View at: Google Scholar
A. Atangana, “A note on the triple Laplace transform and its applications to some kind of third-order differential equation,” Abstract and Applied Analysis, vol. 2013, Article ID 769102, 10 pages, 2013.
View at: Publisher Site | Google Scholar
R. S. Dahiya and M. Vinayagamoorthy, “Laplace transform pairs of n-dimensions and heat conduction problem,” Mathematical and Computer Modelling, vol. 13, no. 10, pp. 35–50, 1990.
View at: Publisher Site | Google Scholar
H. U. Rehman, M. Iftikhar, S. Saleem, M. Younis, and A. Mueed, “A computational quadruple Laplace transform for the solution of partial differential equations,” Applied Mathematics, vol. 5, no. 21, pp. 3372–3382, 2014.
View at: Publisher Site | Google Scholar
S. S. Haider, M. U. Rehman, and T. Abdeljawad, “On Hilfer fractional difference operator,” Advances in Difference Equations, vol. 122, pp. 1–20, 2020.
View at: Google Scholar
S. S. Haider and M. U. Rehman, “On substantial fractional difference operator,” Advances in Difference Equations, no. 1, 2020.
View at: Publisher Site | Google Scholar
M. Bohner, G. S. Guseinov, and B. Karpuz, “Further properties of the Laplace transform on time scales with arbitrary graininess,” Integral Transforms and Special Functions, vol. 24, no. 4, pp. 289–301, 2013.
View at: Publisher Site | Google Scholar
M. D. Ortigueira, F. J. V. Coito, and J. J. Trujillo, “Discrete-time differential systems,” Signal Processing, vol. 107, pp. 198–217, 2015.
View at: Publisher Site | Google Scholar
P. Savoye, “On the inclusion of difference equation problems and Z Transform methods in sophomore differential equation classes,” Primus, vol. 19, no. 6, pp. 491–499, 2009.
View at: Publisher Site | Google Scholar
M. Omran and A. Kiliman, “Fractional double Laplace transform and its properties,” in Proceedings of the AIP Conference, vol. 1795, no. 1, AIP Publishing LLC, Vladivostok, Russia, January 2017.
View at: Google Scholar
J. Peng and K. Li, “A note on property of the Mittag-Leffler function,” Journal of Mathematical Analysis and Applications, vol. 370, no. 2, pp. 635–638, 2010.
View at: Publisher Site | Google Scholar
B. Zhang and Y. Zhou, Qualitative Analysis of Delay Partial Difference Equations, Contemporary Mathematics and its Applications, Hindawi, New York, NY, USA, 2007.
F. Ozpinar and F. M. Belgacem, “The discrete homotopy perturbation Sumudu transform method for solving partial difference equations,” Discrete & Continuous Dynamical Systems, vol. 12, no. 3, pp. 615–624, 2019.
View at: Publisher Site | Google Scholar
A. Aghili and M. R. Masomi, “Solving systems of fractional partial differential equations via two dimensional Laplace transforms,” International Journal of Research–Granthaalayah, vol. 5, no. 12, pp. 406–420, 2017.
View at: Google Scholar
S. K. Elagan, M. Sayed, and M. Higazy, “An analytical study on fractional partial differential equations by Laplace transform operator method,” International Journal of Applied Engineering Research, vol. 13, no. 1, pp. 545–549, 2018.
View at: Google Scholar
A. M. Haghighi and D. P. Mishev, Difference and Differential Equations with Applications in Queueing Theory, John Wiley and Sons, Hoboken, NJ, USA, 2013.
A. M. O. Anwar, F. Jarad, D. Baleaun, and F. Ayaz, “Fractional Caputo heat equation within the double Laplace transform,” Romanian Journal of Physics, vol. 58, pp. 15–18, 2013.
View at: Google Scholar

Copyright

Copyright © 2020 Syed Sabyel Haider et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

480

Downloads

574

Citations