An efficient method for the analytical study of linear and nonlinear time-fractional partial differential equations with variable coefficients

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Abstract

The residual power series method is effective for obtaining approximate analytical solutions to fractional-order differential equations. This method, however, requires the derivative to compute the coefficients of terms in a series solution. Other well-known methods, such as the homotopy perturbation, the Adomian decomposition, and the variational iteration methods, need integration. We are all aware of how difficult it is to calculate the fractional derivative and integration of a function. As a result, the use of the methods mentioned above is somewhat constrained. In this research work, approximate and exact analytical solutions to time-fractional partial differential equations with variable coefficients are obtained using the Laplace residual power series method in the sense of the Gerasimov–Caputo fractional derivative. This method helped us overcome the limitations of the various methods. The Laplace residual power series method performs exceptionally well in computing the coefficients of terms in a series solution by applying the straightforward limit principle at infinity, and it is also more effective than various series solution methods due to the avoidance of Adomian and He polynomials to solve nonlinear problems. The relative, recurrence, and absolute errors of the three problems are investigated in order to evaluate the validity of our method. The results show that the proposed method can be a suitable alternative to the various series solution methods when solving time-fractional partial differential equations.

About the authors

Muhammad Imran Liaqat

Government College University; National College of Business Administration & Economics

Email: imranliaqat50@yahoo.com
ORCID iD: 0000-0002-5732-9689

PhD Student, Abdus Salam School of Mathematical Sciences; Lecturer, Dept. of Mathematics;

Pakistan, 54600, Lahore

Ali Akgül

Lebanese American University; Siirt University; Near East University

Author for correspondence.
Email: aliakgul00727@gmail.com
ORCID iD: 0000-0001-9832-1424

PhD in Math, Full Professor, Dept. of Computer Science and Mathematics; Dept. of Mathematics, Art and Science Faculty; Dept. of Mathematics, Mathematics Research Center

Lebanon, Lebanon, 1102 2801, Beirut; Turkey, 56100, Siirt; Turkey, 99138, Nicosia

Evgenii Yu. Prosviryakov

Ural Federal University; Institute of Engineering Science, RAS (Ural Branch); Urals State University of Railway Transport; Udmurt Federal Research Center, RAS (Ural Branch)

Email: evgen_pros@mail.ru
ORCID iD: 0000-0002-2349-7801

Dr. Phys. & Math. Sci., Dept. of Information Technologies and Control Systems; Sect. of Nonlinear Vortex Hydrodynamics; Dept. of Natural Sciences; Lab. of Physical and Chemical Mechanics

Russian Federation, 620137, Ekaterinburg; 620049, Ekaterinburg; 620034, Ekaterinburg; 426067, Izhevsk

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2. Figure 1. The approximate and exact results of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for various amounts of \(\beta\) in the range \(\omega\in[0,1.0]\), when \(\theta_1=0.5\) (a), \(\theta_1=\theta_2=\theta_3=0.5\) (b), and \(\theta_1=\theta_2=0.5\) (c)

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3. Figure 2. The 2D curves of the Abs-E graph of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for the approximate finding obtained through seven iterations and the exact result in the range \(\omega\in[0,0.5]\), when \(\theta_1=0.25\) (a), \(\theta_1=\theta_2=\theta_3=0.25\) (b), and \(\theta_1=\theta_2=0.25\) (c)

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4. Figure 3. The 2D curves of the Rel-E graph of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for the approximate finding obtained through seven iterations and the exact result in the range \(\omega\in[0,0.5]\), when \(\theta_1=0.25\) (a), \(\theta_1=\theta_2=\theta_3=0.25\) (b), and \(\theta_1=\theta_2=0.25\) (c)

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5. Figure 4. The 3D plots of the Abs-E of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for the approximate finding obtained through seven iterations and exact result in the ranges \(\omega\in[0,0.5]\) and \(\theta_1\in[0,0.5]\), when \(\theta_2=\theta_3=0.25\) (b), and \(\theta_2=0.25\) (c)

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6. Figure 5. The 3D plots of the Rel-E of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for the approximate finding obtained through seven iterations and exact result in the ranges \(\omega\in[0,0.5]\) and \(\theta_1\in[0,0.5]\), when \(\theta_2=\theta_3=0.25\) (b), and \(\theta_2=0.25\) (c)

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