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Original Article

Open Access Peer-reviewed

Daniel Deborah O.^{ }, Ayodele Moyosola

Received September 06, 2020; Revised October 07, 2020; Accepted October 15, 2020

In this paper, the Laplace Differential Transform Method (LDTM) was utilized to solve some nonlinear nonhomogeneous partial differential equations. This technique is the combined form of the Laplace transform method with the Differential Transform Method (DTM). The combined method is efficient in handling nonlinear nonhomogeneous partial differential equations with variable coefficients. Laplace transform is introduced to overcome the inadequacy resulted from unsatisfied boundary condition in using DTM. Illustrative examples were examined to demonstrate the effectiveness of Laplace differential transform method. Results revealed that the LDTM is well appropriate for use in solving such problems.

Many natural events in applied sciences and engineering are modeled with linear and nonlinear partial differential equations as a result of the nature of phenomena happening around the world. Although, most of these problems involving linear or nonlinear partial differential equations do not have analytical solution, hence, numerical or approximate methods have efficiently been used to solve these equations.

Pertubation method ^{ 1} is well known for solving nonlinear partial differential equations but it is only used based on the existence of small parameter. Hence, many methods were presented to eliminate the small parameter, which include: Adominan Decomposition Method ^{ 2, 3}, Variational Iteration Method ^{ 4, 5, 6, 7}, Homotopy Perturbation Method ^{ 8, 9, 10}, Differential transform method ^{ 11, 12, 13, 14}, Homotopy Analysis Method ^{ 15, 16} and Parameter Expansion Method ^{ 17, 18}. But, most of these methods works fine for small range, owning to the satisfied boundary conditions, and the remaining unsatisfied conditions are left out of the final results.

However, in recent research, Laplace transform is being introduced to overcome inadequacy resulted from unsatisfied conditions.

Laplace Homotopy Perturbation Method (LHPM) ^{ 19, 20, 21}, which combines the Laplace transform and Homotopy Pertubation method(HPM), is effectively employed to solve one-dimensional non-homogeneous partial differential equation; Laplace Adominan Decomposition Method (LADM) ^{ 22, 23, 24, 25}, which combines the Laplace transform and Adomian Decomposition Method (ADM), is used for solving nonlinear Volterra integro-differential equations; Laplace Differential transform method (LDTM) ^{ 26}, which combines Laplace transform and Differential Transform Method (DTM), is being used to solve linear non-homogeneous partial differential equations with variable coefficient.

Therefore, in this paper, nonlinear nonhomogeneous partial differential equations with variable coefficient are solved using coupled Laplace transform and Differential Transform Method, with the aim to show the effectiveness of this method in proffering reliable solution to nonlinear PDEs.

In this section, basic idea of Laplace Differential Transform Method (LDTM) is discussed. But, the basic idea of Laplace transform and Differential transform method (DTM) are introduced first before considering the combined form.

Laplace transform is the integral transformation of a given derivative function with real variable t into a complex function with variable s. It is used to convert differential equations into algebraic equations.

The Laplace transform for a function is defined in ^{ 27}

The Laplace transform for a function exist if the transformation integral exist. Therefore,

For some real positive . The integral will converge for , if , the magnitude of is less than i.e. , is the abscissa of absolute convergence.

The basic operations of the dimensional transform which are useful in the transformation in this paper are summarized as follows:

Original FunctionTransformed Function

The differential transform method construct a semi-analytical numerical techniques that makes use of Taylor series for the solution of differential equations in the form of polynomials. ^{ 28}.

The differential transformation of the k th derivative of the function is defined as follows:

(1) |

Where is the original function and is the transformed function. Differential inverse transformation of is defined as follows:

(2) |

The basic operations of the dimensional transform which are useful in the transformation in this paper are summarized as follows: ^{ 28}

Original Function Transformed Function

To illustrate the basic idea of this method, we consider the general form of one-dimensional nonlinear second order nonhomogeneous partial differential equations with variable coefficients of the form:

(3) |

Where is the linear differential operator of order 2, are the variable coefficients, , is the linear operator, is the nonlinear operator and is the source function.

Equation is subject to

(4) |

(5) |

The method involves applying a Laplace transform to equation (3) - (5) and the use of the linearity property of Laplace transform

(6) |

Where is the Laplace transform of .

Applying the initial conditions in equation (4) into equation (6) yields

(7) |

Subject to

(8) |

Which is second order initial value problem.

Accordingly, using differential transform method, the solution of equations (7) and (8) can be written as:

(9) |

Where is the differential transform of and is a function of the parameter. After determining, inverse Laplace transform is applied to equation (9) to get .

Here, the effectiveness of the technique is demonstrated. The advantage of this method is in the ability of applying Laplace transform and differential transform method for obtaining exact solutions of nonlinear nonhomogeneous PDEs. Hence, the effectiveness of the LDTM is demonstrated with the following examples:

Consider the following nonlinear nonhomogeneous equation given by

(10) |

Subject to

(11) |

(12) |

Applying Laplace transform to equation (10) in view of initial condition (11)

(13) |

Subject to

(14) |

Applying differential transform to (13) - (14) yields

(15) |

And

(16) |

Using the recurrence equation (15) and the initial condition (16) yields the following outcomes:

Hence,

(17) |

Inverse Laplace transform of (17) gives

Which is the exact solution to (10) - (12)

Consider the following nonlinear nonhomogeneous equation with variable coefficient given by:

(18) |

Subject to

(19) |

And

(20) |

Applying Laplace transform to equation (18) in view of the initial condition (19)

(21) |

Subject to

(22) |

Applying the differential transform to (21) - (22) yields

(23) |

And

(24) |

By the recurrence equation (23) and the initial conditions (24), we have the following results

Hence,

(25) |

Inverse Laplace transform of equation (25) gives

Which is the exact solution to (18) - (20).

Consider the following nonlinear nonhomogeneous PDE with variable coefficient:

(26) |

Subject to

(27) |

And

(28) |

In lieu of the initial conditions (27), the Laplace transform of (26) gives

(29) |

Subject to

(30) |

Applying differential transform to (29) - (30) gives

(31) |

and

(32) |

By the recurred equation (31) and the initial condition (32) altogether yield

Hence,

(33) |

The inverse Laplace transform of equation (33) yields

Which is the exact solution to (26) - (28).

Consider the following nonlinear nonhomogeneous equation with variable coefficient:

(34) |

Subject to

(35) |

and

(36) |

Laplace transform of equation (34) using (35)

(37) |

Subject to

(38) |

Applying differential transform to equations (37) - (38) gives

(39) |

and

(40) |

By the recurrence equation (39) and the initial condition (40), we have

Hence,

(41) |

The inverse Laplace of equation (41) yields

Which is the exact solution to (34) - (36).

Consider the following nonlinear nonhomogeneous equation with variable coefficient

(42) |

Subject to

(43) |

and

(44) |

For are constants.

Laplace transform of equation (42) using equation (43)

(45) |

Subject to

(46) |

Applying the differential transform to (45) - (46) yield

(47) |

and

(48) |

Where in (47) is the coefficient of the Taylor series. By the recurrence equation (47), and the initial conditions (48), yields

Hence,

(49) |

The inverse Laplace of equation (49) is

The applicability of the combined form of Laplace transform method and Differential Transform Method (DTM) is demonstrated in this paper. This method solves nonlinear nonhomogeneous partial differential equations with variable coefficients. Laplace transform is introduced to overcome the inadequacy resulted from unsatisfied boundary condition using DTM, Consequently, the combined methods efficiently gives the exact solution with little computational work compared to differential transform method. Laplace Differential Transform Method (LDTM) can be applied to solve other nonlinear nonhomogeneous partial differential equations.

[1] | William H. C., Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking, Oxford University Press, 2004, 34. | ||

In article | |||

[2] | Alawneh A., Al-Khaled K, and Al-Towaiq M., Reliable algorithms for solving integro-differential equations with applications, International Journal of Computer Mathematics, 87(7). 1538-1554. 2010. | ||

In article | View Article | ||

[3] | Pukhov G. E., Computational structure for solving differential equations by Taylor transformations. Cybernetics Systems and Analysis, 14. 383. 1978. | ||

In article | View Article | ||

[4] | Alquran M., and Al-Khaled K., Approximate solutions to nonlinear partial integro-differential equations with applications in heat flow, Jordan Journal of Mathematics and Statistics, 3(2). 93-116. 2010. | ||

In article | |||

[5] | Alquran M., and Dogan N., Variational iteration method for solving two-parameter singularly perturbed two point boundary value problem, Applications and Applied Mathematics: An International Journal (AAM), 5(1). 81-95. 2010. | ||

In article | |||

[6] | Ganji D. D., Sadighi A., and Khatami I., Assessment of two analytical approaches in some nonlinear problems arising in engineering science, Physics Letter A, 372. 4399-4406. 2008. | ||

In article | View Article | ||

[7] | Xu L., Variational approach to solution of nonlinear dispersive K(m,n) equation. Chaos Solitons Fractals., 37(1). 137-143. 2008. | ||

In article | View Article | ||

[8] | Alquran M., and Mohammad M., Approximate solutions to system of nonlinear partial differential equations using homotopy perturbation method, International Journal of Nonlinear Science (IJNS), 2(4). 485-497. 2011. | ||

In article | |||

[9] | He J., Homotopy perturbation method for solving boundary value problem. Physics Letter A, 3(5). 87-88. 2006. | ||

In article | View Article | ||

[10] | Moghimi S.M., Ganji D.D., Bararnia H., Hosseini M., and Jalaal M., Homotopy perturbation method for nonlinear MHD Jeffery-Hamel problem, Computer and Mathematics with Application, 61. 2213-2216. 2011. | ||

In article | View Article | ||

[11] | Al-Ahmad., Mamat M., and AlAhmad R., Finding differential transform using difference equation. IAENG International Journal of Applied Mathematics, 50(1). 1-9. 2020. | ||

In article | |||

[12] | Ganji H. F., Jouya M., Mirhosseini-Amiri S. B. and Ganji D. D., Traveling wave solution by differential transformation method and reduced differential transformation method, Alexandria Engineering Journal, 55. 2985-2994. 2016. | ||

In article | View Article | ||

[13] | Ghafoori M., Motevalli M. G., Nejad M. G., Shakeri F., Ganji D. D. and Jalaal M. Efficiency of differential transformation method for nonlinear oscillation: Comparison with HPM and VIM, Current Applied Physics,11. 965-971. 2011. | ||

In article | View Article | ||

[14] | Zou L., Zong Z., Wang Z., and Wang S., Differential transform method for solving solitary wave with discontinuity, Physics Letter A, 374. 3451. 2010. | ||

In article | View Article | ||

[15] | Islam S., Khan Y., Faraz N., and Austin F., Numerical solution of logistic differential equations by using the Laplace decomposition method. World Applied Sciences Journal, 8(1). 1100-1105. 2010. | ||

In article | |||

[16] | Jafari H., Chun C., Seifi S., and Saeidy M., Analytical solution for nonlinear Gas Dynamic equation by Homotopy Analysis Method. Application and Applied Mathematics, 4(1). 149-154. 2009. | ||

In article | |||

[17] | Xu L., Determination of limit cycle by He’s parameter-expanding method for strongly nonlinear oscillators. Journal of Sound and Vibration, 302. 178-184. 2007. | ||

In article | View Article | ||

[18] | Kimiaeifar A., Saidi A. R., Sohouli A. R., and Ganji D. D., Analysis of modified Van der Pol’s oscillator using He’s parameter expanding methods, Current Applied Physics, 10. 279-283. 2010. | ||

In article | View Article | ||

[19] | Madani M., Fathizadeh M., Khan Y., and Yildrim A., On the coupling of the homotopy perturbation method and Laplace transformation, Mathematical and Computer Modelling, 53(9-10). 1937-1945, 2011. | ||

In article | View Article | ||

[20] | Olubanwo O. O., and Odetunde O. S., Laplace homotopy perturbation method of solving nonlinear partial differential equations. Anale. Seria Informatica, Vol XVII Fasc 2. 184-189. 2019. | ||

In article | View Article | ||

[21] | Tripathi R and Mishra H. K., Homotopy perturbation method with Laplace transform (LT-HPM) for solving Lane-Emden type differential equations (LETDEs), SpringerPlus,, 5. 1859. 2016. | ||

In article | View Article PubMed | ||

[22] | Khan, H., Shah, R., Kumam, P., Baleanu D., and Arif M., Laplace decomposition for solving nonlinear system of fractional order partial differential equations, Advances in Difference Equations, 2000. 375. 2020. | ||

In article | View Article | ||

[23] | Purohit M., and Mushtaq S. Applications of Laplace-Adomian decomposition method for solving time-fractional advection dispersion equation. Journal of Mathematical and Computational Science, 10(5). 1960-1968. 2020. | ||

In article | |||

[24] | Shah R., Khan H., Arif M. and Kumam P., Application of Laplace-Adomian decomposition method for the analytical solution of third order dispersive fractional partial differential equations. Entropy, 21(4). 335. 2019. | ||

In article | View Article PubMed | ||

[25] | Wazwaz A., The combined Laplace transform Adomian decomposition method for handling Voltera integrodifferential equations, Applied Mathematics and Computation. 216(4). 1304-1309. 2010. | ||

In article | View Article | ||

[26] | Alquran M., Al-Khaled K., Ali M., and Ta’any A., The combined Laplace transform-differential transform method for solving linear non-homogeneous PDEs, Journal of Mathematical and Computational Science, 2(2). 690-701. 2012. | ||

In article | |||

[27] | Ivanescu M., Mechanical Engineer’s Handbook, Academic Press Series in Engineering, 2003, 611-714. | ||

In article | |||

[28] | Smarda Z., Diblik J. and Khan Y., “Extension of the differential transformation to nonlinear differential and integro-differential equations with proportional delays”, Advance in Difference Equations, 69, 13. 2013. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2020 Daniel Deborah O. and Ayodele Moyosola

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Daniel Deborah O., Ayodele Moyosola. Laplace Differential Transform Method for Solving Nonlinear Nonhomogeneous Partial Differential Equations. *Turkish Journal of Analysis and Number Theory*. Vol. 8, No. 5, 2020, pp 91-96. http://pubs.sciepub.com/tjant/8/5/2

O., Daniel Deborah, and Ayodele Moyosola. "Laplace Differential Transform Method for Solving Nonlinear Nonhomogeneous Partial Differential Equations." *Turkish Journal of Analysis and Number Theory* 8.5 (2020): 91-96.

O., D. D. , & Moyosola, A. (2020). Laplace Differential Transform Method for Solving Nonlinear Nonhomogeneous Partial Differential Equations. *Turkish Journal of Analysis and Number Theory*, *8*(5), 91-96.

O., Daniel Deborah, and Ayodele Moyosola. "Laplace Differential Transform Method for Solving Nonlinear Nonhomogeneous Partial Differential Equations." *Turkish Journal of Analysis and Number Theory* 8, no. 5 (2020): 91-96.

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[1] | William H. C., Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking, Oxford University Press, 2004, 34. | ||

In article | |||

[2] | Alawneh A., Al-Khaled K, and Al-Towaiq M., Reliable algorithms for solving integro-differential equations with applications, International Journal of Computer Mathematics, 87(7). 1538-1554. 2010. | ||

In article | View Article | ||

[3] | Pukhov G. E., Computational structure for solving differential equations by Taylor transformations. Cybernetics Systems and Analysis, 14. 383. 1978. | ||

In article | View Article | ||

[4] | Alquran M., and Al-Khaled K., Approximate solutions to nonlinear partial integro-differential equations with applications in heat flow, Jordan Journal of Mathematics and Statistics, 3(2). 93-116. 2010. | ||

In article | |||

[5] | Alquran M., and Dogan N., Variational iteration method for solving two-parameter singularly perturbed two point boundary value problem, Applications and Applied Mathematics: An International Journal (AAM), 5(1). 81-95. 2010. | ||

In article | |||

[6] | Ganji D. D., Sadighi A., and Khatami I., Assessment of two analytical approaches in some nonlinear problems arising in engineering science, Physics Letter A, 372. 4399-4406. 2008. | ||

In article | View Article | ||

[7] | Xu L., Variational approach to solution of nonlinear dispersive K(m,n) equation. Chaos Solitons Fractals., 37(1). 137-143. 2008. | ||

In article | View Article | ||

[8] | Alquran M., and Mohammad M., Approximate solutions to system of nonlinear partial differential equations using homotopy perturbation method, International Journal of Nonlinear Science (IJNS), 2(4). 485-497. 2011. | ||

In article | |||

[9] | He J., Homotopy perturbation method for solving boundary value problem. Physics Letter A, 3(5). 87-88. 2006. | ||

In article | View Article | ||

[10] | Moghimi S.M., Ganji D.D., Bararnia H., Hosseini M., and Jalaal M., Homotopy perturbation method for nonlinear MHD Jeffery-Hamel problem, Computer and Mathematics with Application, 61. 2213-2216. 2011. | ||

In article | View Article | ||

[11] | Al-Ahmad., Mamat M., and AlAhmad R., Finding differential transform using difference equation. IAENG International Journal of Applied Mathematics, 50(1). 1-9. 2020. | ||

In article | |||

[12] | Ganji H. F., Jouya M., Mirhosseini-Amiri S. B. and Ganji D. D., Traveling wave solution by differential transformation method and reduced differential transformation method, Alexandria Engineering Journal, 55. 2985-2994. 2016. | ||

In article | View Article | ||

[13] | Ghafoori M., Motevalli M. G., Nejad M. G., Shakeri F., Ganji D. D. and Jalaal M. Efficiency of differential transformation method for nonlinear oscillation: Comparison with HPM and VIM, Current Applied Physics,11. 965-971. 2011. | ||

In article | View Article | ||

[14] | Zou L., Zong Z., Wang Z., and Wang S., Differential transform method for solving solitary wave with discontinuity, Physics Letter A, 374. 3451. 2010. | ||

In article | View Article | ||

[15] | Islam S., Khan Y., Faraz N., and Austin F., Numerical solution of logistic differential equations by using the Laplace decomposition method. World Applied Sciences Journal, 8(1). 1100-1105. 2010. | ||

In article | |||

[16] | Jafari H., Chun C., Seifi S., and Saeidy M., Analytical solution for nonlinear Gas Dynamic equation by Homotopy Analysis Method. Application and Applied Mathematics, 4(1). 149-154. 2009. | ||

In article | |||

[17] | Xu L., Determination of limit cycle by He’s parameter-expanding method for strongly nonlinear oscillators. Journal of Sound and Vibration, 302. 178-184. 2007. | ||

In article | View Article | ||

[18] | Kimiaeifar A., Saidi A. R., Sohouli A. R., and Ganji D. D., Analysis of modified Van der Pol’s oscillator using He’s parameter expanding methods, Current Applied Physics, 10. 279-283. 2010. | ||

In article | View Article | ||

[19] | Madani M., Fathizadeh M., Khan Y., and Yildrim A., On the coupling of the homotopy perturbation method and Laplace transformation, Mathematical and Computer Modelling, 53(9-10). 1937-1945, 2011. | ||

In article | View Article | ||

[20] | Olubanwo O. O., and Odetunde O. S., Laplace homotopy perturbation method of solving nonlinear partial differential equations. Anale. Seria Informatica, Vol XVII Fasc 2. 184-189. 2019. | ||

In article | View Article | ||

[21] | Tripathi R and Mishra H. K., Homotopy perturbation method with Laplace transform (LT-HPM) for solving Lane-Emden type differential equations (LETDEs), SpringerPlus,, 5. 1859. 2016. | ||

In article | View Article PubMed | ||

[22] | Khan, H., Shah, R., Kumam, P., Baleanu D., and Arif M., Laplace decomposition for solving nonlinear system of fractional order partial differential equations, Advances in Difference Equations, 2000. 375. 2020. | ||

In article | View Article | ||

[23] | Purohit M., and Mushtaq S. Applications of Laplace-Adomian decomposition method for solving time-fractional advection dispersion equation. Journal of Mathematical and Computational Science, 10(5). 1960-1968. 2020. | ||

In article | |||

[24] | Shah R., Khan H., Arif M. and Kumam P., Application of Laplace-Adomian decomposition method for the analytical solution of third order dispersive fractional partial differential equations. Entropy, 21(4). 335. 2019. | ||

In article | View Article PubMed | ||

[25] | Wazwaz A., The combined Laplace transform Adomian decomposition method for handling Voltera integrodifferential equations, Applied Mathematics and Computation. 216(4). 1304-1309. 2010. | ||

In article | View Article | ||

[26] | Alquran M., Al-Khaled K., Ali M., and Ta’any A., The combined Laplace transform-differential transform method for solving linear non-homogeneous PDEs, Journal of Mathematical and Computational Science, 2(2). 690-701. 2012. | ||

In article | |||

[27] | Ivanescu M., Mechanical Engineer’s Handbook, Academic Press Series in Engineering, 2003, 611-714. | ||

In article | |||

[28] | Smarda Z., Diblik J. and Khan Y., “Extension of the differential transformation to nonlinear differential and integro-differential equations with proportional delays”, Advance in Difference Equations, 69, 13. 2013. | ||

In article | View Article | ||