The Solutions of Partial Differential Equations and Recent Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (6 March 2022) | Viewed by 20970

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Guest Editor
Department of Mathematics and Statistics, Universiti Putra Malaysia, Serdang, 43400 Selangor, Malaysia
Interests: integral transforms and special functions; generalized functions; generalized hypergeometric functions; distributions; ultra-distributions; topological semigroups, fractional integro-differential equations; fractals and fractional inequalities; fuzzy soft sets and applications in decision making
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Special Issue Information

Dear Colleagues,

Partial differential equations with fractional and integer orders have been applied in many modeling problems. They are inspired by problems which arise in diverse fields such as biology, finance, physics, differential geometry, control theory, as well as engineering. Furthermore, the formulations of theorems that describe the initial value and boundary value problems are of particular interest to PDE. Here, we consider the wide range of applications, such as some physical applications—in particular, the fractional form of the advection–dispersion equation—and the fundamental solution in the form of the Lévy α-stable distribution density among the important applications of fractional modeling. In this Special Issue, we aim to cover the recent development of the typical models, to generalize the known standard problems, and to replace classical terms with fractional forms.

Dr. Adem Kilicman
Guest Editor

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Keywords

  • linear and quasi-linear PDEs
  • classification of PDEs
  • Ill-posed and well-posed problems
  • dirichlet and Neumann problems in PDEs
  • maximum principles
  • fractional modeling and PDEs

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Published Papers (10 papers)

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Research

20 pages, 343 KiB  
Article
On the Application of Multi-Dimensional Laplace Decomposition Method for Solving Singular Fractional Pseudo-Hyperbolic Equations
by Hassan Eltayeb, Adem Kılıçman and Imed Bachar
Fractal Fract. 2022, 6(11), 690; https://doi.org/10.3390/fractalfract6110690 - 21 Nov 2022
Cited by 1 | Viewed by 1267
Abstract
In this work, the exact and approximate solution for generalized linear, nonlinear, and coupled systems of fractional singular M-dimensional pseudo-hyperbolic equations are examined by using the multi-dimensional Laplace Adomian decomposition method (M-DLADM). In particular, some two-dimensional illustrative examples are provided to confirm the [...] Read more.
In this work, the exact and approximate solution for generalized linear, nonlinear, and coupled systems of fractional singular M-dimensional pseudo-hyperbolic equations are examined by using the multi-dimensional Laplace Adomian decomposition method (M-DLADM). In particular, some two-dimensional illustrative examples are provided to confirm the efficiency and accuracy of the present method. Full article
16 pages, 371 KiB  
Article
A Unified Inertial Iterative Approach for General Quasi Variational Inequality with Application
by Mohammad Akram and Mohammad Dilshad
Fractal Fract. 2022, 6(7), 395; https://doi.org/10.3390/fractalfract6070395 - 18 Jul 2022
Cited by 2 | Viewed by 1409
Abstract
In this paper, we design two inertial iterative methods involving one and two inertial steps for investigating a general quasi-variational inequality in a real Hilbert space. We establish an existence result and a non-trivial example is furnished to substantiate our theoretical findings. We [...] Read more.
In this paper, we design two inertial iterative methods involving one and two inertial steps for investigating a general quasi-variational inequality in a real Hilbert space. We establish an existence result and a non-trivial example is furnished to substantiate our theoretical findings. We discuss the convergence of the inertial iterative algorithms to approximate the solution of a general quasi-variational inequality. Finally, we apply an inertial iterative scheme with two inertial steps to investigate a delay differential equation. The results presented herein can be seen as substantial generalizations of some known results. Full article
12 pages, 1827 KiB  
Article
Financial Applications on Fractional Lévy Stochastic Processes
by Reem Abdullah Aljethi and Adem Kılıçman
Fractal Fract. 2022, 6(5), 278; https://doi.org/10.3390/fractalfract6050278 - 22 May 2022
Cited by 3 | Viewed by 1924
Abstract
In this present work, we perform a numerical analysis of the value of the European style options as well as a sensitivity analysis for the option price with respect to some parameters of the model when the underlying price process is driven by [...] Read more.
In this present work, we perform a numerical analysis of the value of the European style options as well as a sensitivity analysis for the option price with respect to some parameters of the model when the underlying price process is driven by a fractional Lévy process. The option price is given by a deterministic representation by means of a real valued function satisfying some fractional PDE. The numerical scheme of the fractional PDE is obtained by means of a weighted and shifted Grunwald approximation. Full article
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13 pages, 389 KiB  
Article
New Explicit Solutions of the Extended Double (2+1)-Dimensional Sine-Gorden Equation and Its Time Fractional Form
by Gangwei Wang, Li Li, Qi Wang and Juan Geng
Fractal Fract. 2022, 6(3), 166; https://doi.org/10.3390/fractalfract6030166 - 17 Mar 2022
Cited by 1 | Viewed by 1666
Abstract
In this paper, the extended double (2+1)-dimensional sine-Gorden equation is studied. First of all, using the symmetry method, the corresponding vector fields, Lie algebra and infinitesimal generators are derived. Then, from infinitesimal generators, the symmetry reductions are presented. In addition, these reduced equations [...] Read more.
In this paper, the extended double (2+1)-dimensional sine-Gorden equation is studied. First of all, using the symmetry method, the corresponding vector fields, Lie algebra and infinitesimal generators are derived. Then, from infinitesimal generators, the symmetry reductions are presented. In addition, these reduced equations are converted into the corresponding partial differential equations, which including classical double (1+1)-dimensional sine-Gorden equation. Moreover, based on the Lie symmetry method again, these reduced equations are investigated. Meanwhile, based on traveling wave transformation, some explicit solutions of the extended double (2+1)-dimensional sine-Gorden equation are obtained. Consequently, a conservation law is derived via conservation law multiplier method. Finally, especially with the help of the fractional complex transform, some solutions of double time fractional (2+1)-dimensional sine-Gorden equation are also derived. These results might explain complex nonlinear phenomenon. Full article
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16 pages, 4224 KiB  
Article
Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method
by Adel R. Hadhoud, Abdulqawi A. M. Rageh and Taha Radwan
Fractal Fract. 2022, 6(3), 127; https://doi.org/10.3390/fractalfract6030127 - 23 Feb 2022
Cited by 11 | Viewed by 2007
Abstract
This paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo sense. The [...] Read more.
This paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo sense. The L1approximation method is used for time-fractional derivative discretization. Using Von Neumann stability analysis, the proposed technique is shown to be conditionally stable. Numerical examples are solved to verify the accuracy and effectiveness of this method. The error norms L2 and L are also calculated at different values of N and t to evaluate the performance of the suggested method. Full article
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12 pages, 332 KiB  
Article
Results on Neutral Partial Integrodifferential Equations Using Monch-Krasnosel’Skii Fixed Point Theorem with Nonlocal Conditions
by Chokkalingam Ravichandran, Kasilingam Munusamy, Kottakkaran Sooppy Nisar and Natarajan Valliammal
Fractal Fract. 2022, 6(2), 75; https://doi.org/10.3390/fractalfract6020075 - 31 Jan 2022
Cited by 38 | Viewed by 2343
Abstract
In this theory, the existence of a mild solution for a neutral partial integrodifferential nonlocal system with finite delay is presented and proved using the techniques of the Monch–Krasnosel’skii type of fixed point theorem, a measure of noncompactness and resolvent operator theory. For [...] Read more.
In this theory, the existence of a mild solution for a neutral partial integrodifferential nonlocal system with finite delay is presented and proved using the techniques of the Monch–Krasnosel’skii type of fixed point theorem, a measure of noncompactness and resolvent operator theory. For this work, we have introduced some sufficient conditions to confirm the existence of the neutral partial integrodifferential system. An illustration of the derived results is offered at the end with a filter system corresponding to our existence result. Full article
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28 pages, 596 KiB  
Article
Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects
by Ridhwan Reyaz, Ahmad Qushairi Mohamad, Yeou Jiann Lim, Muhammad Saqib and Sharidan Shafie
Fractal Fract. 2022, 6(1), 38; https://doi.org/10.3390/fractalfract6010038 - 12 Jan 2022
Cited by 14 | Viewed by 2375
Abstract
Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations [...] Read more.
Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations for fractional derivatives in the mechanics of fluid flows are yet to be discovered. Nonetheless, theoretical studies will be useful in facilitating future experimental studies. Therefore, the aim of this study is to showcase an analytical solution on the impact of the Caputo-Fabrizio fractional derivative for a magnethohydrodynamic (MHD) Casson fluid flow with thermal radiation and chemical reaction. Analytical solutions are obtained via Laplace transform through compound functions. The obtained solutions are first verified, then analysed. It is observed from the study that variations in the fractional derivative parameter, α, exhibits a transitional behaviour of fluid between unsteady state and steady state. Numerical analyses on skin friction, Nusselt number, and Sherwood number were also analysed. Behaviour of these three properties were in agreement of that from past literature. Full article
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13 pages, 348 KiB  
Article
The Darboux Transformation and N-Soliton Solutions of Coupled Cubic-Quintic Nonlinear Schrödinger Equation on a Time-Space Scale
by Huanhe Dong, Chunming Wei, Yong Zhang, Mingshuo Liu and Yong Fang
Fractal Fract. 2022, 6(1), 12; https://doi.org/10.3390/fractalfract6010012 - 29 Dec 2021
Cited by 4 | Viewed by 1421
Abstract
The coupled cubic-quintic nonlinear Schrödinger (CQNLS) equation is a universal mathematical model describing many physical situations, such as nonlinear optics and Bose–Einstein condensate. In this paper, in order to simplify the process of similar analysis with different forms of the coupled CQNLS equation, [...] Read more.
The coupled cubic-quintic nonlinear Schrödinger (CQNLS) equation is a universal mathematical model describing many physical situations, such as nonlinear optics and Bose–Einstein condensate. In this paper, in order to simplify the process of similar analysis with different forms of the coupled CQNLS equation, this dynamic system is extended to a time-space scale based on the Lax pair and zero curvature equation. Furthermore, Darboux transformation of the coupled CQNLS dynamic system on a time-space scale is constructed, and the N-soliton solution is obtained. These results effectively combine the theory of differential equations with difference equations and become a bridge connecting continuous and discrete analysis. Full article
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17 pages, 814 KiB  
Article
Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition
by Serik E. Aitzhanov, Abdumauvlen S. Berdyshev and Kymbat S. Bekenayeva
Fractal Fract. 2021, 5(4), 134; https://doi.org/10.3390/fractalfract5040134 - 23 Sep 2021
Cited by 5 | Viewed by 1537
Abstract
This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in [...] Read more.
This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in the form of a nonlinear boundary condition with a fractional differentiation operator. The main result of this work is establishing the local or global solvability of stated problems, depending on the parameters of the equation. The Galerkin method is used to prove the existence of a quasi-linear pseudo-parabolic equation’s weak solution in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. A priori estimates and the Rellich–Kondrashov theorem are used to prove the existence of the desired solutions to the considered boundary value problems. The uniqueness of the weak generalized solutions of the initial boundary value problems is proved on the basis of the obtained a priori estimates and the application of the generalized Gronwall lemma. The need to consider and study such initial boundary value problems for a quasi-linear pseudo-parabolic equation follows from practical requirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc. Full article
22 pages, 11613 KiB  
Article
Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations
by Supaporn Kaewta, Sekson Sirisubtawee, Sanoe Koonprasert and Surattana Sungnul
Fractal Fract. 2021, 5(3), 88; https://doi.org/10.3390/fractalfract5030088 - 4 Aug 2021
Cited by 11 | Viewed by 2558
Abstract
The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable [...] Read more.
The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations. Full article
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