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Journal articles on the topic 'Two-dimensional differential transform; Partial differential equations; Differential transform method'

Author: Grafiati

Published: 5 June 2025

Last updated: 15 July 2025

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1

Fadwa, A. M. Madi, and Abdelwahid Fawzi. "A NEW APPROACH ON THE TWO-DIMENSIONAL DIFFERENTIAL TRANSFORM." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 10, no. 08 (2022): 2855–59. https://doi.org/10.5281/zenodo.7014640.

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In recent years, new formulas of the two-dimensional differential transform have been proven by using the definition of the transform.  In this work, we use a new approach based on the definition of the transform and the summation properties to prove the two-dimensional differential transform of the product of two functions, then we used this result to establish other useful formulas.  This study shows that this procedure can be used to find formulas for many complicated terms.  This enables us to apply the differential transform method on many types of partial differential equa

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2

Firat, Omer, and Ozan OZKAN. "NEW TWO DIMENSIONAL DIFFERENTIAL TRANSFORM METHOD WITH CONFORMABLE DERIVATIVE FOR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS." Journal of Inequalities and Special Functions 13, no. 4 (2022): 12–24. https://doi.org/10.54379/jiasf-2022-4-2.

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In this paper, the method of differential transform with two variables is combined the newly defined conformable derivative for fractional calculus. Some useful fundamental properties of presented method are given and used to get solutions of linear and non-linear differential problems with fractional order. To control the reliability, applicability and correctness of the technique we applied the method to distinct fractional partial differential problems which have exact solutions in the literature. The obtained results show that this suggested method is useful, appropriate and practical to l

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3

Kuang Chen, Cha'o, and Shing Huei Ho. "Solving partial differential equations by two-dimensional differential transform method." Applied Mathematics and Computation 106, no. 2-3 (1999): 171–79. http://dx.doi.org/10.1016/s0096-3003(98)10115-7.

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4

Mikaeilvand, N., and S. Khakrangin. "Solving fuzzy partial differential equations by fuzzy two-dimensional differential transform method." Neural Computing and Applications 21, S1 (2012): 307–12. http://dx.doi.org/10.1007/s00521-012-0901-x.

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5

Chen, Xue Hui, Liang Wei, Lian Cun Zheng, and Xin Xin Zhang. "Analytical Approach to Time-Fractional Partial Differential Equations in Fluid Mechanics." Advanced Materials Research 347-353 (October 2011): 463–66. http://dx.doi.org/10.4028/www.scientific.net/amr.347-353.463.

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The generalized differential transform method is implemented for solving time-fractional partial differential equations in fluid mechanics. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor’s formula. Results obtained by using the scheme presented here agree well with the numerical results presented elsewhere. The results reveal the method is feasible and convenient for handling approximate solutions of time-fractional partial differential equations.

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Chen, Xue Hui, Liang Wei, Ji Zhe Sui, and Lian Cun Zheng. "Solving the Linear Time-Fractional Wave Equation by Generalized Differential Transform Method." Applied Mechanics and Materials 204-208 (October 2012): 4476–80. http://dx.doi.org/10.4028/www.scientific.net/amm.204-208.4476.

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In this paper, the generalized differential transform method is implemented for solving time-fractional wave equations in fluid mechanics. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor’s formula. The results reveal the method is feasible and convenient for handling approximate solutions of time-fractional partial differential equations.

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Haydar, Amal Khalaf, Radhi A. Zaboon, and Shatha S. Alhily. "Solving nth order non-homogenous complex partial differential equation using two dimensional differential transform method." Journal of Interdisciplinary Mathematics 28, no. 4 (2025): 1385–96. https://doi.org/10.47974/jim-1879.

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This paper utilizes two-dimensional differential transformation for some complex two-variables functions. A recurrence formula related to a complex partial differential equation of order n is presented. In addition, 2-dimensional differential transformation method is used to introduce analytical solutions to complex partial differential equations by the obtained recurrence relation.

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8

Benhammouda, Brahim, Hector Vazquez-Leal, and Arturo Sarmiento-Reyes. "Modified Reduced Differential Transform Method for Partial Differential-Algebraic Equations." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/279481.

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This work presents the application of the reduced differential transform method (RDTM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-two and index-three are solved to show that RDTM can provide analytical solutions for PDAEs in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful technique to find exact solutions. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend

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9

Alhassan, Amal, Radhi Ali Zaboon, and Shatha Alhily. "Solving Higher Orders Linear Complex Partial Differential Equations via Two Dimensional Differential Transform Method." Journal of University of Anbar for Pure Science 18, no. 1 (2024): 257–62. http://dx.doi.org/10.37652/juaps.2023.144285.1156.

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Kularathna, R. S. M., N. Kajan, T. Jeyamugan, and S. Thilaganathan. "Solution of Laplace Equation by Modified Differential Transform Method." Asian Research Journal of Mathematics 20, no. 3 (2024): 51–58. http://dx.doi.org/10.9734/arjom/2024/v20i3790.

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In this paper, we applied the modified two-dimensional differential transform method to solve Laplace equation. Laplace equation is one of Elliptic partial differential equations. These kinds of differential equations have specific applications models of physics and engineering. We consider four models with two Dirichlet and two Neumann boundary conditions. The simplicity of this method compared to other iteration methods is shown here. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.

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11

Yu, Jianping, Jian Jing, Yongli Sun, and Suping Wu. "(n+1)-Dimensional reduced differential transform method for solving partial differential equations." Applied Mathematics and Computation 273 (January 2016): 697–705. http://dx.doi.org/10.1016/j.amc.2015.10.016.

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12

Merdan, Mehmet, Merve Merdan, and Rıdvan Şahin. "Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations." Cumhuriyet Science Journal 45, no. 3 (2024): 562–70. http://dx.doi.org/10.17776/csj.1256101.

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[1] M. Duz, Solution of complex differential equations with variable coefficients by using reduced differential transform, Miskolc Mathematical Notes Vol. 21 (2020), No. 1, pp. 161–170 [2] M. Duz, “Application of Elzaki Transform to first order constant coefficients complex equa ions.” Bulletin of the international mathematical virtual institute., vol. 7, pp. 387–393, 2017, doi: 10.7251/BIMVI1702387D. [3] M. Duz, “On an application of Laplace transforms.” ¨ NTMSCI., vol. 5, no. 2, pp. 193–198, 2017, doi: 10.12732/ijam.v31i1.2. [4] M. Duz, “Solution of complex equations with Adomian Decompositi

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13

Bildik, Necdet, and Ali Konuralp. "Two-dimensional differential transform method, Adomian's decomposition method, and variational iteration method for partial differential equations." International Journal of Computer Mathematics 83, no. 12 (2006): 973–87. http://dx.doi.org/10.1080/00207160601173407.

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14

Baleanu, Dumitru, and Hassan Kamil Jassim. "Exact Solution of Two-Dimensional Fractional Partial Differential Equations." Fractal and Fractional 4, no. 2 (2020): 21. http://dx.doi.org/10.3390/fractalfract4020021.

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In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations.

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Issa, Ahmad, and Emad A. Kuffi. "On The Double Integral Transform (Complex EE Transform) and Their Properties and Applications." Ibn AL-Haitham Journal For Pure and Applied Sciences 37, no. 1 (2024): 429–41. http://dx.doi.org/10.30526/37.1.3329.

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Due to the importance of solutions of partial differential equations, linear, nonlinear, homogeneous, and non-homogeneous, in important life applications, including engineering applications, physics and astronomy, medical sciences, and life technology, and their importance in solutions to heat transfer equations, wave, Laplace equation, telegraph, etc. In this paper, a new double integral transform has been proposed. In this work, we have introduced a new double transform ( Double Complex EE Transform ). In addition, we presented the convolution theorem and proved the properties of the propose

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Sahraee, Zahra, and Maryam Arabameri. "A Semi-Discretization Method Based on Finite Difference and Differential Transform Methods to Solve the Time-Fractional Telegraph Equation." Symmetry 15, no. 9 (2023): 1759. http://dx.doi.org/10.3390/sym15091759.

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The telegraph equation is a hyperbolic partial differential equation that has many applications in symmetric and asymmetric problems. In this paper, the solution of the time-fractional telegraph equation is obtained using a hybrid method. The numerical simulation is performed based on a combination of the finite difference and differential transform methods, such that at first, the equation is semi-discretized along the spatial ordinate, and then the resulting system of ordinary differential equations is solved using the fractional differential transform method. This hybrid technique is tested

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D.O., Anongo, and Awari Y.S. "Solution of One-dimensional Partial Differential Equation with Higher-Order Derivative by Double Laplace Transform Method." African Journal of Mathematics and Statistics Studies 4, no. 3 (2021): 1–11. http://dx.doi.org/10.52589/ajmss-1ohgjpnr.

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Many problems in natural and engineering sciences such as heat transfer, elasticity, quantum mechanics, water flow, and others are modelled mathematically by partial differential equations. Some of these problems may be linear, nonlinear, homogeneous, non-homogeneous, and order greater or equal one. Finding the theoretical solution to these problems with less cumbersome techniques is an active area of research in the aforementioned field. In this research paper, we have developed a new application of the double Laplace transform method to solve homogeneous and non-homogeneous linear partial di

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18

Osman, Mawia, Almegdad Almahi, Omer Abdalrhman Omer, Altyeb Mohammed Mustafa, and Sarmad A. Altaie. "Approximation Solution for Fuzzy Fractional-Order Partial Differential Equations." Fractal and Fractional 6, no. 11 (2022): 646. http://dx.doi.org/10.3390/fractalfract6110646.

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In this article, the authors study the comparison of the generalization differential transform method (DTM) and fuzzy variational iteration method (VIM) applied to determining the approximate analytic solutions of fuzzy fractional KdV, K(2,2) and mKdV equations. Furthermore, we establish the approximation solution two-and three-dimensional fuzzy time-fractional telegraphic equations via the fuzzy reduced differential transform method (RDTM). Finding an exact or closed-approximation solution to a differential equation is possible via fuzzy RDTM. Finally, we present the fuzzy fractional variatio

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19

Zedan, Hassan A., and M. Ali Alghamdi. "Solution of (3+1)-Dimensional Nonlinear Cubic Schrodinger Equation by Differential Transform Method." Mathematical Problems in Engineering 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/531823.

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Four-dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time. Moreover, as an application of four-dimensional differential transform, exact solutions of nonlinear system of partial differential equations have been investigated. The results of the present method are compared very well with analytical solution of the system. Differential transform method can easily be applied to linear or nonlinear problems and reduces the size of computational work. With this method, exact solutions may be obtained without any need of cumbersom

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20

Deresse, Alemayehu Tamirie. "Application of Iterative Three-Dimensional Laplace Transform Method for 2-Dimensional Nonlinear Klein-Gordon Equation." Trends in Sciences 20, no. 3 (2023): 4410. http://dx.doi.org/10.48048/tis.2023.4410.

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In the present study, the exact analytical solutions of the 2-dimensional nonlinear Klein-Gordon equation (NLKGE) is investigated using the 3-dimensional Laplace transform method in conjunction with the Daftardar-Gejji and Jafari Method (Iterative method). Through this method, the linear part of the problem is solved by using the 3-dimensional Laplace transform method, while the noise terms from the nonlinear part of the equation disappear through a successive iteration process of the Daftardar-Gejji and Jafari Method (DJM), where a single iteration gives the exact solution to the problem. Thr

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Zhu, Dexiang, Shikun Dai, Qingrui Chen, and Hongjun Tian. "Three-dimension holographic numerical simulation of gravity anomalies." Journal of Physics: Conference Series 2718, no. 1 (2024): 012047. http://dx.doi.org/10.1088/1742-6596/2718/1/012047.

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Abstract In gravity-anomaly-based prospecting, it is difficult to achieve large-scale and high-precision inversion imaging of complex geological models. To address this issue, this paper proposes a three-dimensional holographic numerical simulation method for gravity anomalies. This method transforms the 3D partial differential equation of gravity potential into many independent 1D differential equations with different wavenumbers by performing a 2D Fourier transform along the horizontal direction. It decomposes a large-scale 3D numerical modeling problem into many 1D numerical modeling proble

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22

H. Al-Tai, Marwa, and Ali Al-Fayadh. "Solving Two Dimensional Coupled Burger's Equations and Sine-Gordon Equation Using El-Zaki Transform-Variational Iteration Method." Al-Nahrain Journal of Science 24, no. 2 (2021): 41–47. http://dx.doi.org/10.22401/anjs.24.2.07.

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In this paper, the combined form of the Elzaki transform and variation iteration method is implemented efficiently in finding the analytical and numerical solutions of the two-dimensional nonlinear coupled Burger's partial differential equations and sine-Gordon partial differential equation. The obtained solutions were compared to the exact solutions and other existing methods. Illustrative examples show the efficiency and the power of the used method.

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23

Mirzaee, Farshid, and Mohammad Komak Yari. "A novel computing three-dimensional differential transform method for solving fuzzy partial differential equations." Ain Shams Engineering Journal 7, no. 2 (2016): 695–708. http://dx.doi.org/10.1016/j.asej.2015.05.013.

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24

Arshad, Muhammad, Dianchen Lu, and Jun Wang. "( N +1)-dimensional fractional reduced differential transform method for fractional order partial differential equations." Communications in Nonlinear Science and Numerical Simulation 48 (July 2017): 509–19. http://dx.doi.org/10.1016/j.cnsns.2017.01.018.

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25

Bagyalakshmi, Morachan, and G. SaiSundarakrishnan. "Tarig Projected Differential Transform Method to solve fractional nonlinear partial differential equations." Boletim da Sociedade Paranaense de Matemática 38, no. 3 (2019): 23–46. http://dx.doi.org/10.5269/bspm.v38i3.34432.

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Recent advancement in the field of nonlinear analysis and fractional calculus help to address the rising challenges in the solution of nonlinear fractional partial differential equations. This paper presents a hybrid technique, a combination of Tarig transform and Projected Differential Transform Method (TPDTM) to solve nonlinear fractional partial differential equations. The effectiveness of the method is examined by solving three numerical examples that arise in the field of heat transfer analysis. In this proposed scheme, the solution is obtained as a convergent series and the result is use

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26

Al-Saif, A. S. J., and Zinah A. Hasan. "An analytical approximate method for solving unsteady state two-dimensional convection-diffusion equations." JOURNAL OF ADVANCES IN MATHEMATICS 21 (June 22, 2022): 73–88. http://dx.doi.org/10.24297/jam.v21i.9242.

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In this paper, an analytic approximate method for solving the unsteady two-dimensional convection-diffusion equations is introduced. Also, the convergence of the approximate methods is analyzed. Three test examples are presented, two have exact and one has not exacted solutions. The results obtained show that these methods are powerful mathematical tools for solving linear and nonlinear partial differential equations, moreover, new analytic Taylor method (NATM), reduced differential transform method (RDTM), and homotopy perturbation method (HPM), are more accurate and have less CPU time than t

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Mtawal, Ahmad. "Application of the Sumudu Variational Iteration Method with Atangana-Baleanu-Caputo Operator for Solving Fractional-Order Heat-Like Equations with Initial Conditions." Journal of Pure & Applied Sciences 23, no. 2 (2024): 50–60. http://dx.doi.org/10.51984/jopas.v23i2.3151.

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Fractional calculus techniques are widely utilized across various engineering disciplines and applied sciences. Among these techniques is the Sumudu Variational Iteration Method (SVIM), which has not yet been tested with the Atangana-Baleanu-Caputo fractional derivative in academic literature. This work aims to explore the application of SVIM for solving fractional-order partial differential equations using the Atangana-Baleanu-Caputo derivative. The method integrates the Sumudu transform with the variational iteration method. To demonstrate the effectiveness and validity of SVIM, we apply it

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SUN, JIANSHE. "ANALYTICAL APPROXIMATE SOLUTIONS OF (N + 1)-DIMENSIONAL FRACTAL HARRY DYM EQUATIONS." Fractals 26, no. 06 (2018): 1850094. http://dx.doi.org/10.1142/s0218348x18500949.

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The new fractal models of the [Formula: see text]-dimensional and [Formula: see text]-dimensional nonlinear local fractional Harry Dym equation (HDE) on Cantor sets are derived and the analytical approximate solutions of the above two new models are obtained by coupling the fractional complex transform via local fractional derivative (LFD) and local fractional reduced differential transform method (LFRDTM). Fractional complex transform for functions of [Formula: see text]-dimensional variables is generalized and the theorems of [Formula: see text]-dimensional LFRDTM are supplementary extended.

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Et al., Enadi. "New Approach for Solving Three Dimensional Space Partial Differential Equation." Baghdad Science Journal 16, no. 3(Suppl.) (2019): 0786. http://dx.doi.org/10.21123/bsj.2019.16.3(suppl.).0786.

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This paper presents a new transform method to solve partial differential equations, for finding suitable accurate solutions in a wider domain. It can be used to solve the problems without resorting to the frequency domain. The new transform is combined with the homotopy perturbation method in order to solve three dimensional second order partial differential equations with initial condition, and the convergence of the solution to the exact form is proved. The implementation of the suggested method demonstrates the usefulness in finding exact solutions. The practical implications show the effec

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30

Chen, Qingrui, Shuqing Ma, Lilun Zhang, Shikun Dai, and Jiaxuan Ling. "Three-dimensional modeling of Marine Controlled Source Electromagnetic field using a space-wavenumber domain method." Journal of Physics: Conference Series 2718, no. 1 (2024): 012055. http://dx.doi.org/10.1088/1742-6596/2718/1/012055.

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Abstract Under complex conditions, the three-dimensional modeling of Marine Controlled Source Electromagnetic field requires a significant amount of computation, resulting in slow calculation speed and high storage requirements. To solve these problems, we propose a 3D numerical simulation method of electromagnetic field in the space-wavenumber domain under the Lorenz gauge. Firstly, the new method utilizes the two-dimensional Fourier transform in the horizontal direction to transform the 3D partial differential equations of the Lorenz vector potentials into multiple independent ordinary diffe

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Jacobs, B. A., and C. Harley. "Two Hybrid Methods for Solving Two-Dimensional Linear Time-Fractional Partial Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/757204.

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A computationally efficient hybridization of the Laplace transform with two spatial discretization techniques is investigated for numerical solutions of time-fractional linear partial differential equations in two space variables. The Chebyshev collocation method is compared with the standard finite difference spatial discretization and the absolute error is obtained for several test problems. Accurate numerical solutions are achieved in the Chebyshev collocation method subject to both Dirichlet and Neumann boundary conditions. The solution obtained by these hybrid methods allows for the evalu

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LÜ, XING, TAO GENG, CHENG ZHANG, HONG-WU ZHU, XIANG-HUA MENG, and BO TIAN. "MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION." International Journal of Modern Physics B 23, no. 25 (2009): 5003–15. http://dx.doi.org/10.1142/s0217979209053382.

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In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneous

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Zhang, Tao, and Xinran Zhu. "A Solution Method for Partial Differential Equations by Fusing Fourier Operators and U-Net." Mathematics 13, no. 7 (2025): 1033. https://doi.org/10.3390/math13071033.

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In scientific and engineering calculation, the effective solution of partial differential equations (PDEs) has great significance. This paper presents an innovative method based on the combination of a U-Net neural network with Fourier neural operators, aiming to improve the accuracy and efficiency of solving partial differential equations. U-Net neural networks with a unique encoding–decoder structure and hopping connections can efficiently extract and integrate spatial-domain features and accurately describe the spatial structure of PDEs. The Fourier neural operator combines the advantages o

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Eltayeb, Hassan. "Application of the Double Sumudu-Generalized Laplace Transform Decomposition Method to Solve Singular Pseudo-Hyperbolic Equations." Symmetry 15, no. 9 (2023): 1706. http://dx.doi.org/10.3390/sym15091706.

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In this study, the technique established by the double Sumudu transform in combination with a new generalized Laplace transform decomposition method, which is called the double Sumudu-generalized Laplace transform decomposition method, is applied to solve general two-dimensional singular pseudo-hyperbolic equations subject to the initial conditions. The applicability of the proposed method is analyzed through demonstrative examples. The results obtained show that the procedure is easy to carry out and precise when employed for different linear and non-linear partial differential equations.

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Abdallah, Habila Ali, and Yousif Elriyah Abrar. "Using Adomian decomposition methods for solving systems of nonlinear partial differential equations." International Journal of Mathematics and Physical Sciences Research 10, no. 2 (2022): 14–22. https://doi.org/10.5281/zenodo.7182781.

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<strong>Abstract:</strong> In this paper, we apply the Adomian decomposition method (ADM) and Modified decomposition method (MDM) on two different types of nonlinear partial differential equations (PDEs), has been solved by using the homotopy perturbation method combined with new transform (NTHPM). But after solved by (MADM) we found (MADM) has less of computational work than (NTHPM), more effective, powerful and simple than (NTHPM). <strong>Keywords:</strong> Systems of nonlinear partial differential equations, Adomian decomposition method, Modified decomposition method, homotopy perturbation

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Gusu, Daba Meshesha, Dechasa Wegi, Girma Gemechu, and Diriba Gemechu. "Fractional Order Airy’s Type Differential Equations of Its Models Using RDTM." Mathematical Problems in Engineering 2021 (September 10, 2021): 1–21. http://dx.doi.org/10.1155/2021/3719206.

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In this paper, we propose a novel reduced differential transform method (RDTM) to compute analytical and semianalytical approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions. The performance of the proposed method was analyzed and compared with a convergent series solution form with easily computable coefficients. The behavior of approximated series solutions at different values of fractional order α and its modeling in 2-dimensional and 3-dimensional sp

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Hadhoud, Adel R., Abdulqawi A. M. Rageh, and Taha Radwan. "Employing the Laplace Residual Power Series Method to Solve (11)- and (21)-Dimensional Time-Fractional Nonlinear Differential Equations++." Fractal and Fractional 8, no. 7 (2024): 401. http://dx.doi.org/10.3390/fractalfract8070401.

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In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional partial differential equations. Then, we apply the proposed method to find approximate solutions to the time-fractional coupled Berger equations, the time-fractional coupled Korteweg–de Vries equations and time-fractional Whitham–Broer–Kaup equations. Secondly, we extend the proposed method to so

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Seo, Jeong-Kweon, and Byeong-Chun Shin. "Reduced-order modeling using the frequency-domain method for parabolic partial differential equations." AIMS Mathematics 8, no. 7 (2023): 15255–68. http://dx.doi.org/10.3934/math.2023779.

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<abstract><p>This paper suggests reduced-order modeling using the Galerkin proper orthogonal decomposition (POD) to find approximate solutions for parabolic partial differential equations. We first transform a parabolic partial differential equation to the frequency-dependent elliptic equations using the Fourier integral transform in time. Such a frequency-domain method enables efficiently implementing a parallel computation to approximate the solutions because the frequency-variable elliptic equations have independent frequencies. Then, we introduce reduced-order modeling to deter

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Zhang, Feng, Yuru Hu, and Xiangpeng Xin. "Lie Symmetry Analysis, Exact Solutions, and Conservation Laws of Variable-Coefficients Boiti-Leon-Pempinelli Equation." Advances in Mathematical Physics 2021 (November 29, 2021): 1–14. http://dx.doi.org/10.1155/2021/6227384.

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In this article, we study the generalized ( 2 + 1 )-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optim

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Saadeh, Rania. "Application of the ARA Method in Solving Integro-Differential Equations in Two Dimensions." Computation 11, no. 1 (2022): 4. http://dx.doi.org/10.3390/computation11010004.

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The main purpose of this study is to investigate solutions of some integral equations of different classes using a new scheme. This research introduces and implements the new double ARA transform to solve integral and partial integro-differential equations. We introduce basic theorems and properties of the double ARA transform in two dimensions, and some results related to the double convolution theorem and partial derivatives are presented. In addition, to show the validity of the proposed technique, we introduce and solve some examples using the new approach.

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Li, Wei, Jianxin Liu, Rongwen Guo, and Hang Ji. "Two-Dimensional Magnetotelluric Forward Modeling in Space-wavenumber Mixed Domain." Journal of Physics: Conference Series 2651, no. 1 (2023): 012069. http://dx.doi.org/10.1088/1742-6596/2651/1/012069.

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Abstract In magnetotelluric exploration, if you want to do 2D or 3D numerical modelling, what you need in the computational and memory requirements are potentially enormous. In order to get efficient and precise forward simulation results, we propose a new 2D numerical modelling method in space-wavenumber mixed domain. This method uses the one-dimensional Fourier transform in the horizontal directions, so the two-dimensional partial differential equations in the spatial domain can be transformed into a set of independent one-dimensional differential equations with different wave numbers. Then

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Osman, Mawia, Yonghui Xia, Muhammad Marwan, and Omer Abdalrhman Omer. "Novel Approaches for Solving Fuzzy Fractional Partial Differential Equations." Fractal and Fractional 6, no. 11 (2022): 656. http://dx.doi.org/10.3390/fractalfract6110656.

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In this paper, we present a comparison of several important methods to solve fuzzy partial differential equations (PDEs). These methods include the fuzzy reduced differential transform method (RDTM), fuzzy Adomian decomposition method (ADM), fuzzy Homotopy perturbation method (HPM), and fuzzy Homotopy analysis method (HAM). A distinguishing practical feature of these techniques is administered without the need to use discretion or restricted assumptions. Moreover, we investigate the fuzzy (n+1)-dimensional fractional RDTM to obtain the solutions of fuzzy fractional PDEs. The much more distinct

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Sumiati, Ira, and Sukono Sukono. "Adomian Decomposition Method and The Other Integral Transform." Operations Research: International Conference Series 1, no. 4 (2020): 110–13. http://dx.doi.org/10.47194/orics.v1i4.151.

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The Adomian decomposition method is an iterative method that can be used to solve integral, differential, and integrodifferential equations. The differential equations that can be solved by this method can be of integer or fractional order, ordinary or partial, with initial or boundary value problems, with variable or constant coefficients, linear or nonlinear, homogeneous or nonhomogeneous. This method divides the equation into two forms, namely linear and nonlinear, so that it can solve equations without linearization, discretization, perturbation, or other restrictive assumptions. The basic

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Sharma, Dinkar, Gurpinder Singh Samra, and Prince Singh. "Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method." Nonlinear Engineering 9, no. 1 (2020): 370–81. http://dx.doi.org/10.1515/nleng-2020-0023.

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AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

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Abdelrahman, Mahmoud A. E. "A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations." Nonlinear Engineering 7, no. 4 (2018): 279–85. http://dx.doi.org/10.1515/nleng-2017-0145.

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AbstractIn this paper, the fractional derivatives in the sense of modified Riemann–Liouville and the Riccati-Bernoulli Sub-ODE method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Zoomeron equation and the (3 + 1) dimensional space-time fractional mKDV-ZK equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The Riccati-Ber

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Khalid, Aliyu Muhammad1 2. Aliyu Aliyu Isa2 3. Tiwari Sarita1 2. Sylvain Meinrad Donkeng Voumo 4*. "Application of Reduced Differential Transform Method to Solve Linear, Non-Linear Convection-Diffusion and Reaction-Diffusion Problems." MSI Journal of Multidisciplinary Research (MSIJMR) Volume 2, Issue 5 (2025): 55–64. https://doi.org/10.5281/zenodo.15372549.

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Convection-diffusion and reaction-diffusion equations are fundamental in describing various physical phenomena, yet their solution, particularly for nonlinear cases, often presents significant mathematical challenges. This study investigates the application of the Reduced Differential Transform Method (RDTM) to obtain analytical solutions for both linear and nonlinear convection-diffusion and reaction-diffusion problems. The RDTM, derived from power series expansion, was systematically applied to four illustrative examples: two linear convection-diffusion equations and two nonlinear reaction-d

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Eltayeb, Hassan. "Analytic Solution of the Time-Fractional Partial Differential Equation Using a Multi-G-Laplace Transform Method." Fractal and Fractional 8, no. 8 (2024): 435. http://dx.doi.org/10.3390/fractalfract8080435.

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In several recent studies, many researchers have shown the advantage of fractional calculus in the production of particular solutions of a huge number of linear and nonlinear partial differential equations. In this research work, different theorems related to the G-double Laplace transform (DGLT) are proved. The solution of the system of time-fractional partial differential equations is addressed using a new analytical method. This technique is a combination of the multi-G-Laplace transform and decomposition methods (MGLTDM). Moreover, we discuss the convergence of this method. Two examples ar

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Salman, Zahrah I., Majid Tavassoli Kajani, Mohammed Sahib Mechee, and Masoud Allame. "Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs." Mathematics 11, no. 17 (2023): 3786. http://dx.doi.org/10.3390/math11173786.

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Proposing a matrix transform method to solve a fractional partial differential equation is the main aim of this paper. The main model can be transferred to a partial-integro differential equation (PIDE) with a weakly singular kernel. The spatial direction is approximated by a fourth-order difference scheme. Also, the temporal derivative is discretized via a second-order numerical procedure. First, the spatial derivatives are approximated by a fourth-order operator to compute the second-order derivatives. This process produces a system of differential equations related to the time variable. The

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LI, WU, and TIAN YOU FAN. "STUDY ON ELASTIC ANALYSIS OF CRACK PROBLEM OF TWO-DIMENSIONAL DECAGONAL QUASICRYSTALS OF POINT GROUP 10, $\overline {10}$." Modern Physics Letters B 23, no. 16 (2009): 1989–99. http://dx.doi.org/10.1142/s0217984909020151.

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By introducing a stress potential function, we transform the plane elasticity equations of two-dimensional quasicrystals of point group 10, [Formula: see text] to a partial differential equation. And then we use the complex variable function method for classical elasticity theory to that of the quasicrystals. As an example, a decagonal quasicrystal in which there is an arc is subjected to a uniform pressure p in the elliptic notch of the decagonal quasicrystal is considered. With the help of conformal mapping, we obtain the exact solution for the elliptic notch problem of quasicrystals. The wo

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Parmeshwari Aland. "Time-Fractional Hyperbolic Telegraph Equation: A Semi-Analytic Approach Using Modified Adomian Decomposition Elzaki Transform Method." Communications on Applied Nonlinear Analysis 32, no. 4s (2024): 309–31. https://doi.org/10.52783/cana.v32.2815.

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In this research paper, an approximate analytical solution approach known as the Modified Adomian Decomposition Method with the coupling of Elzaki Transform (MADETM) is deployed for addressing one-dimensional, two- dimensional, and three-dimensional time-fractional hyperbolic telegraph equations. The Caputo derivative operator yields the approximate analytical solution. The impact ness and its accuracy of the adopted method are demonstrated through comparison of the approximate results with the exact solutions, both presented graphically by plotting its surface graph, line graph through analyz

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