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A novel image denoising model, namely Full Fractional Total Variation (TVFF), based on the Rudin-Osher-Fatemi (ROF) and the fractional total variation models is presented. The leading advantage of TVFF model is that it uses fractional derivatives with length scale parameters instead of ordinary derivatives with respect to both time and spatial variables in the diffusion equation. The Riesz–Caputo fractional derivative operator is used to disperse nonlocal influence throughout all directions, whereas the Caputo fractional derivative concept is employed for time fractional derivatives. Therefore, the influence of neighboring pixels is given greater weight compared to those situated farther away and this reflects the consideration behind denoising process better. Moreover, the numerical approach is constructed, and its stability and convergence properties are thoroughly examined. To show the superiority of our model, the denoised images are subjected to visual and numerical comparisons using metrics such as the Signal-to-Noise Ratio (SNR), the Structural Similarity Index Measure (SSIM) and the Edge-Retention Ratio (ERR). The performance of the TVFF method is evaluated under various types of noise, including Poisson, Speckle, and Salt & Pepper, and the results are compared with those obtained using Gauss and Median Filters. Furthermore, the proposed method is applied to both blind and synthetic images, thereby showcasing its versatility and applicability across diverse datasets. The outcomes showcase the substantial potential of our enhanced model as a versatile and efficient tool for image denoising. [ABSTRACT FROM AUTHOR]

The primary aim of this study is to analyze the chaotic dynamics of a conformable maturity structured cell partial differential equation of order z ∈ (0 , 1) , which extends the classical Lasota equation. To examine the chaotic behavior of our model’s solution, we initially extend certain criteria of linear chaos to conformable calculus. This extension is crucial because the solution of our model does not generate a classical semigroup but rather a c 0 -z-semigroup. For the velocity term of our model, B (w) = μ w , where μ ∈ C , and the term source g (w , ϑ (r , w)) , we utilize spectral properties of the z-infinitesimal generator to demonstrate chaotic behavior in the space C (J 0 , C) , J 0 : = [ 0 , + ∞) . Furthermore, by employing conformable admissible weight functions and setting B (w) = 1 , we establish chaos in the solution z-semigroup, this time within the space C 0 (J 0 , C) . [ABSTRACT FROM AUTHOR]

In this paper, the approximate solutions of the time fractional Harry Dym equation with fractional derivative in the Caputo sense are obtained by using the Residual power series method (RPSM). This equation is a significant dynamical equation that occurs in a variety of physical systems. The suggested method provides good accuracy for the approximate solution when compared numerically with the exact solution. The effectiveness of the proposed method is also illustrated with the aid of numerical results. These results indicate that the RPSM is a power, useful, and applicable for determining the solutions of the time Hary Dym equation. Some of these results are illustrated by 2D and 3D graphics. Besides, the proposed method can be applied to many different differential equations due to its ease of use and reliability. [ABSTRACT FROM AUTHOR]

Abstract Wazwaz Kaur Boussinesq (WKB) equation can effectively simulate the behavior of water waves in shallow water, including the nonlinear effect and dispersion phenomenon of waves, which is of great significance for understanding the dynamic process of ocean, river and other water bodies. To enrich the wave equation theory, the (3+1)-dimensional integer order derivative of WKB equation is changed to the fractional one with beta derivative. The current work deals with the fractional (3+1)-dimensional WKB equation for discussing its chaotic behavior and establishing some new analytic solutions. The chaotic properties of the equation are verified by the trend of evolution along with time, Lyapunov exponents and initial sensitivity analysis. And then complete discrimination system for polynomial method is applied to derive some trigonometric, hyperbolic, Jacobi elliptic and other solutions. The graphical demonstrations are provided for part of these solutions. From these visualized graphs, the solitary, periodic and quasi-periodic wave are shown and the effect of fractional derivatives on the equation can be seen intuitively.

This article presents a range of new exact travelling wave solutions for the nonlinear time-fractional Clannish Random Walker’s Parabolic equation, utilising trigonometric and hyperbolic functions in the context of the beta derivative. This approach enables the effective modelling of non-local and memory effects in wave propagation. The proposed analytical methods significantly enhance computational efficiency, reliability, and effectiveness, providing a diverse set of exact travelling wave solutions. Modified auxiliary equation and the generalised projective Riccati equation methods are employed to extract travelling solitary wave solutions with high amplitude, unique periodic, dark and bright solitons, high amplitude lump wave, breather wave, kink wave, and anti-kink with multiple lump wave profiles. Using the modern software Maple, graphs of solitary travelling wave solutions in three-dimensional, contour, and two-dimensional formats are presented, effectively demonstrating the parameters’ impact. To observe dynamic insight, this study presents sensitivity and chaotic analysis. This study helps to predict lump and high amplitude soliton, showing stable and sharp transitions often happen in fluid dynamics and fibre optics. [ABSTRACT FROM AUTHOR]

Mathematics, mathematical modeling of real systems, and mathematical and computer methodologies aimed at the qualitative and quantitative study of real physical systems interact in a nontrivial way. This work aims to examine a new class of inverse problems for a fractional partial differential equation with order fractional 0<ρ≤1$$ 0<\rho \le 1 $$, which leads to the spectral problem involving Hermite's differential equation. We introduce proven theorems on the existence and uniqueness of solutions to the current problem. We obtain solutions in the form of series expansion using the Hermite orthogonal basis. Finally, we discuss the convergence analysis of the obtained solutions. [ABSTRACT FROM AUTHOR]

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 are provided to check the applicability and efficiency of our technique. [ABSTRACT FROM AUTHOR]

Abstract It is mentioned that understanding linear and non-linear thermo-elasticity systems is important for understanding temperature, elasticity, stresses, and thermal conductivity. One of the most crucial aspects of the current research is the solution to these systems. The fractional form of several thermo-elastic systems is explored, and elegant solutions are provided. The solutions of fractional and integer thermo-elastic systems are further discussed using tables and diagrams. The closed contact between the LRPSM and exact solutions is displayed in the graphs. Plotting fractional problem solutions demonstrates their convergence towards integer-order problem solutions for suitable modeling. The tables confirm that greater precision is rapidly attained as the terms of the derived series solution increase. The faster convergence and stability of the suggested method support its modification for other fractional non-linear complex systems in nature.

It is mentioned that understanding linear and non-linear thermo-elasticity systems is important for understanding temperature, elasticity, stresses, and thermal conductivity. One of the most crucial aspects of the current research is the solution to these systems. The fractional form of several thermo-elastic systems is explored, and elegant solutions are provided. The solutions of fractional and integer thermo-elastic systems are further discussed using tables and diagrams. The closed contact between the LRPSM and exact solutions is displayed in the graphs. Plotting fractional problem solutions demonstrates their convergence towards integer-order problem solutions for suitable modeling. The tables confirm that greater precision is rapidly attained as the terms of the derived series solution increase. The faster convergence and stability of the suggested method support its modification for other fractional non-linear complex systems in nature. [ABSTRACT FROM AUTHOR]

The paper proposes a fractional-differential modification of the mathematical model of the process of nonstationary charging of polar dielectric materials under conditions of irradiation with medium-energy electron beams. The mathematical formalization is based on a spherically symmetric diffusion–drift equation with a fractional time derivative. An implicit finite-difference scheme is constructed using the Caputo derivative approximation. An application program has been developed in Matlab software that implements the designed computational algorithm. Verification of an approximate solution of the problem is demonstrated using a test example. The results of computational experiments to evaluate the characteristics of field effects of injected charges in ferroelectrics when varying the order of fractional differentiation in subdiffusion regimes are presented. [ABSTRACT FROM AUTHOR]

In this paper, using Sinc-Galerkin method and TSVD regularization, an approximation of the quasi-solution to an inverse source problem is obtained. To do so, the solution of direct problem is obtained by the Sinc-Galerkin method, and this solution is applied in a least squares cost functional. Then, to obtain an approximation of the quasi-solution, we minimize the cost functional by TSVD regularization. Error analysis and convergence of the proposed method are investigated. In addition, at the end, four numerical examples are given in details to show the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Lie symmetry analysis is considered as one of the most powerful techniques that has been used for analyzing and extracting various types of solutions to partial differential equations. Conservation laws reflect important aspects of the behavior and properties of physical systems. This paper focuses on the investigation of the (1+1)-dimensional time-fractional modified Benjamin-Bona-Mahony equation (mBBM) incorporating Riemann-Liouville derivatives (RLD). Through the application of Lie symmetry analysis, the study explores similarity reductions and transforms the problem into a nonlinear ordinary differential equation with fractional order. A power series solution is obtained using the Erdelyi-Kober fractional operator, and the convergence of the solutions is analyzed. Furthermore, novel conservation laws for the time-fractional mBBM equation are established. The findings of the current work contribute to a deeper understanding of the dynamics of this fractional evolution equation and provide valuable insights into its behavior. [ABSTRACT FROM AUTHOR]

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 are provided to check the applicability and efficiency of our technique.

This paper studies symmetry reductions of time fractional K(m,1) equation with generalized evolution which is one of the important equation in fractional partial differential equations (FPDEs). Here, the classical Lie symmetries are obtained and utilized for the reduction of time fractional K(m,1) equation into ordinary differential equations. Some new exact solutions are obtained which are in the form of solitary waves by using G′G-expansion method and one of the obtained solutions is acceptable for qualitative analysis. By showing the phase portraits, its dynamical behaviour is further completely explained.

In the present work, the fractional‐order Sawada–Kotera–Ito problem is solved by considering nonlocal Caputo and nonsingular Atangana–Baleanu (ABC) derivatives. The methodology used is an application of the Shehu transform and the Adomian decomposition method. The obtained solution is more accurate when using the ABC type derivative as compared to the Caputo sense, when using the proposed ADShTM method (Adomian decomposition Shehu transform method). The results so obtained by the ADShTM using Caputo and ABC operators are compared, establishing the superiority of the proposed method. The numerical results demonstrate that the application of the ABC derivative is not only relatively more effective and reliable but also straightforward to achieve high precision solution. [ABSTRACT FROM AUTHOR]

The present study introduces and explores the novel application of the Fractional Residual Power Series Method as a highly efficient and reliable approach for solving the challenging non-linear, time-fractional Kawahara and modified Kawahara equations. These equations, which are fifth-order, non-linear partial differential equations, are of significant importance in modeling shallow water waves. In contrast to existing methods such as the Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), and Adomian Decomposition Method (ADM), our research uniquely demon- strates the superior performance of the Fractional Residual Power Series Method. We emphasize that our approach not only outperforms these conventional techniques but also offers ease of implementation. This novel methodology promises to advance the field of mathematical modeling, providing a powerful tool for solving complex problems in the field of science and engineering. [ABSTRACT FROM AUTHOR]

In this research paper, the third-order fractional partial differential equation (FPDE) in the sense of the Caputo fractional derivative and the Atangana-Baleanu Caputo (ABC) fractional derivative is investigated for the first time. The importance of the proposed problem is more general than the third-order linear time-varying systems model. This fractional partial differential equation is converted to the abstract form of the ordinary differential equation at β = 1. A first-order finite difference scheme is created for the converted problem. The stability inequality of the (FDS) is demonstrated for this abstract form problem using Cardano's relation method in Hilbert space. The explicit finite difference method (EFDM) is used to obtain an approximate solution of the third-order fractional partial differential equation (FPDE) based on the Caputo fractional derivative and the (ABC) fractional derivative of order β∈(0,1]. Concerning these equations, the obtained approximate solutions using the proposed method are contrasted with the exact solution. Finally, the simulation for the third-order (FPDE) based on the fractional derivatives of Caputo and (ABC) for both the exact and approximate solutions is given.

In this paper, we deal with the multiterm time‐fractional partial differential equation involving the Caputo operator associated with the Laplace operator, which includes the momentum equations of the fractional Oldroyd‐B fluid and the fractional Burgers fluid under some suitable conditions. By using several properties of multivariate Mittag–Leffler functions, the well‐posedness and the long‐time behavior for the Dirichlet problem are obtained. In addition, the uniqueness in inverse problem of determining orders of time‐fractional derivatives of the equation is proved. [ABSTRACT FROM AUTHOR]

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization. [ABSTRACT FROM AUTHOR]

Initial-boundary value problems including space-time fractional PDEs have been used to model a wide range of problems in physics and engineering fields. In this paper, a non-self adjoint initial boundary value problem containing a third order fractional differential equation is considered. First, a spectral problem for this problem is presented. Then the eigenvalues and eigenfunctions of the main spectral problem are calculated. In order to calculate the roots of their characteristic equation, the asymptotic expansion of the roots is used. Finally, by suitable choice of these asymptotic expansions, related eigenfunctions and Mittag-Lefler functions, the analytic and numerical solutions to the main initial-boundary value problem are given.

In this paper, we study the forced oscillation of solutions of a fractional partial differential system with damping terms by using the Riemann-Liouville derivative and integral. We obtained some new oscillation results by using the integral averaging technique. The obtained results are illustrated by using some examples.

In this paper, we consider a class of fractional partial differential equations with damping term subject to Robin and Dirichlet boundary value conditions. We derive several new sufficient criteria for oscillation of solutions of the equations by using the integral averaging technique and generalized Riccati type transformations. Some applications of the main results are illustrated by some examples.

The key objective of the current manuscript was to investigate the exact solutions of the fractional perturbed Radhakrishnan–Kundu–Lakshmanan model. For this purpose, we applied two reliable and efficient approaches; specifically, the modified simple equation (MSE) and exponential rational function (ERF) techniques. The methods considered in this paper offer solutions for problems in nonlinear theory and mathematical physics practice. We also present solutions obtained graphically with the Maple package program. [ABSTRACT FROM AUTHOR]

The nonlinearity in many problems occurs because of the complexity of the given physical phenomena. The present paper investigates the non-linear fractional partial differential equations’ solutions using the Caputo operator with Laplace residual power series method. It is found that the present technique has a direct and simple implementation to solve the targeted problems. The comparison of the obtained solutions has been done with actual solutions to the problems. The fractional-order solutions are presented and considered to be the focal point of this research article. The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem. Because of the simple implementation, the present technique can be extended to solve other important fractional order problems. [ABSTRACT FROM AUTHOR]

The main objective of the paper is to study the non‐local problem for a pseudo‐parabolic equation with fractional time and space. The derivative of time is understood in the sense of the time derivative of the Caputo fraction of the order α, 0 < α < 1. The first result is an investigation of the existence and uniformity of the solution; the formula for mild solution and the regularity properties will be given. The proofs are based on a number of sophisticated techniques using the Sobolev embedding and also on the construction of the Mittag–Lefler operator. In the second part, we investigate the convergence of the mild solution for non‐local problem to the solution of the local problem when two non‐local parameters reach 0. Finally, we present some numerical examples to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

This paper is to investigate the extent and speed of the spread of the coronavirus disease 2019 (COVID-19) pandemic in the United States (US). For this purpose, the fractional form of the susceptible-exposed-infected-recovered-vaccinated-quarantined-hospitalized-social distancing (SEIR-VQHP) model is initially developed, considering the effects of social distancing, quarantine, hospitalization, and vaccination. Then, a Monte Carlo-based back analysis method is proposed by defining the model parameters, viz. the effects of social distancing rate (α), infection rate (β), vaccination rate (ρ), average latency period (γ), infection-to-quarantine rate (δ), time-dependent recovery rate (λ), time-dependent mortality rate (κ), hospitalization rate (ξ), hospitalization-to-recovery rate (ψ), hospitalization-to-mortality rate (ϕ), and the fractional degree of differential equations as random variables, to obtain the optimal parameters and provide the best combination of fractional order so as to give the best possible fit to the data selected between January 20, 2020 and February 10, 2021. The results demonstrate that the number of infected, recovered, and dead cases by the end of 2021 will reach 1.0, 49.8, and 0.7 million, respectively. Moreover, the histograms of the fractional order acquired from back analysis are provided that can be utilized in similar fractional analyses as an informed initial suggestion. Furthermore, a sensitivity analysis is provided to investigate the effect of vaccination and social distancing on the number of infected cases. The results show that if the social distancing increases by 25% and the vaccination rate doubles, the number of infected cases will drop to 0.13 million by early 2022, indicating relative pandemic control in the US. [ABSTRACT FROM AUTHOR]

In this research paper, the third-order fractional partial differential equation (FPDE) in the sense of the Caputo fractional derivative and the Atangana-Baleanu Caputo (ABC) fractional derivative is investigated for the first time. The importance of the proposed problem is more general than the third-order linear time-varying systems model. This fractional partial differential equation is converted to the abstract form of the ordinary differential equation at β = 1. A first-order finite difference scheme is created for the converted problem. The stability inequality of the (FDS) is demonstrated for this abstract form problem using Cardano's relation method in Hilbert space. The explicit finite difference method (EFDM) is used to obtain an approximate solution of the third-order fractional partial differential equation (FPDE) based on the Caputo fractional derivative and the (ABC) fractional derivative of order β∈(0,1]. Concerning these equations, the obtained approximate solutions using the proposed method are contrasted with the exact solution. Finally, the simulation for the third-order (FPDE) based on the fractional derivatives of Caputo and (ABC) for both the exact and approximate solutions is given. [ABSTRACT FROM AUTHOR]

Human face recognition has the most noteworthy role in detecting a person in many real-world scenarios in computer vision like identification, authentication, security, and so on. Face recognition typically acquires the features and compares them to a dataset to discover the best match. The existing methods failed to accurately extract the robust features for face recognition. To solve these issues, Gaussian Diffusive Hartigan Multidimensional Deep Belief Divergence Feature Learning (GDH-MDBDFL) method is proposed based on the Fractional Partial Differential Equation (FPDE) for face recognition. The proposed GDH-MDBDFL method is designed to improve the accuracy of face recognition using FPDE. The proposed GDH-MDBDFL method comprises three different layers such as input, output, and four hidden layers. Both qualitative and quantitative result analysis is presented to verify the effectiveness of the proposed GDH-MDBDFL method. The simulation results, the proposed GDH-MDBDFL method gives the higher recognition accuracy, and precision with lesser recognition time compared to the conventional methods. [ABSTRACT FROM AUTHOR]

In these analyses, we consider the time-fractional Fisher equation in two-dimensional space. Through the use of the Riemann-Liouville derivative approach, the well-known Lie point symmetries of the utilized equation are derived. Herein, we overturn the fractional fisher model to a fractional differential equation of nonlinear type by considering its Lie point symmetries. The diminutive equation's derivative is in the Erdélyi-Kober sense, whilst we use the technique of the power series to conclude explicit solutions for the diminutive equations for the first time. The conservation laws for the dominant equation are built using a novel conservation theorem. Several graphical countenances were utilized to award a visual performance of the obtained solutions. Finally, some concluding remarks and future recommendations are utilized.

In this paper, we study the forced oscillation of solutions of a fractional partial differential system with damping terms by using the Riemann-Liouville derivative and integral. We obtained some new oscillation results by using the integral averaging technique. The obtained results are illustrated by using some examples. [ABSTRACT FROM AUTHOR]

In this research paper, the authors aim to establish a novel algorithm in the finite difference method (FDM). The novel idea is proposed in the mesh generation process, the process to generate random grids. The FDM over a randomly generated grid enables fast convergence and improves the accuracy of the solution for a given problem; it also enhances the quality of precision by minimizing the error. The FDM involves uniform grids, which are commonly used in solving the partial differential equation (PDE) and the fractional partial differential equation. However, it requires a higher number of iterations to reach convergence. In addition, there is still no definite principle for the discretization of the model to generate the mesh. The newly proposed method, which is the SM method, employed randomly generated grids for mesh generation. This method is compared with the uniform grid method to check the validity and potential in minimizing the computational time and error. The comparative study is conducted for the first time by generating meshes of different cell sizes, i.e., 10×10,20×20,30×30,40×4010\times 10,\hspace{.25em}20\times 20,\hspace{.25em}30\times 30,\hspace{.25em}40\times 40 using MATLAB and ANSYS programs. The two-dimensional PDEs are solved over uniform and random grids. A significant reduction in the computational time is also noticed. Thus, this method is recommended to be used in solving the PDEs.

In this paper, we investigate a BVP for a class impulsive fractional partial differential equations. We propose a new topological approach to prove the existence of at least one classical solution and at least two nonnegative classical solutions. The arguments are based upon recent theoretical results. [ABSTRACT FROM AUTHOR]

This article deals with a semilinear time-space fractional diffusion equation and a coupled semilinear diffusion system of two time-space fractional diffusion equations in the whole Euclidean space. The time-fractional derivative used here is in the sense of ψ -Caputo derivative with order in (0, 1), and the space derivative is the so-called fractional Laplacian with order also in (0, 1). The Cauchy problem of the semilinear equation is firstly studied by constructing its mild solution and weak solution where its mild solution is shown to be also its weak solution. The finite-time blow-up and global existence of the solution to this semilinear equation are studied. Following almost the same procedure, the blow-up in a finite time and global existence of the solution to the coupled semilinear system are investigated too. Finally, the finite-time blow-up is tested by numerical simulations. [ABSTRACT FROM AUTHOR]

The Cauchy problem of a time–space fractional partial differential equation which has as a particular case the damped wave equation is solved for the Dirac delta initial condition. The solution is obtained in terms of H-Fox functions and models the travel of a disturbance in a vibrating medium. [ABSTRACT FROM AUTHOR]

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization.

We investigate a boundary value problem for partial differential equation with Cauchy-Euler fractional operator x 2 α D ∗ , x 2 α u (x , y) + x α D ∗ , x α u (x , y) + u yy (x , y) = f (x , y) , 0 < x < ∞ , 0 < y < 1 , 0 < α < 1. The limitations on convergence of series and the Mellin transform lead to the search for solutions in the class of functions, which possess continuous fractional derivatives for x ∈ (0 , 1) and equal to zero for x > 1 . However, for some cases, the solution is found for the entire semi-infinite strip x > 0 . The analysis is supported by the computations, which exhibit the high level of accuracy. [ABSTRACT FROM AUTHOR]

The solutions to fractional differentials equations are very difficult to investigate. In particular, the solutions of fractional partial differential equations are challenging tasks for mathematicians. In the present article, an extension to this idea is presented to obtain the solutions of non-linear fractional Korteweg–de Vries equations. The solutions comparison of the proposed problems is done via two analytical procedures, which are known as the Residual power series method (RPSM) and q-HATM, respectively. The graphical and tabular analysis are presented to show the reliability and competency of the suggested techniques. The comparison has shown the greater contact between exact, RPSM, and q-HATM solutions. The fractional solutions are in good control and provide many important dynamics of the given problems.

The key objective of the current manuscript was to investigate the exact solutions of the fractional perturbed Radhakrishnan–Kundu–Lakshmanan model. For this purpose, we applied two reliable and efficient approaches; specifically, the modified simple equation (MSE) and exponential rational function (ERF) techniques. The methods considered in this paper offer solutions for problems in nonlinear theory and mathematical physics practice. We also present solutions obtained graphically with the Maple package program.

In this paper, we are interested in the fractional Yamabe-type equation A s u = u n + 2 s n - 2 s , u > 0 in Ω and u = 0 on ∂ Ω. Here Ω is a regular bounded domain of R n , n ≥ 2 and A s , s ∈ (0 , 1) represents the fractional Laplacian operator in Ω with zero Dirichlet boundary condition. Based on the theory of critical points at infinity of Bahri and the localization technique of Caffarelli and Silvestre, we compute the difference of topology induced by the critical points at infinity between the level sets of the variational functional associated to the problem. Our result can be seen as a nonlocal analog of the theorem of Bahri et al. (Cal. Var. Partial. Differ. Equ. 3 (1995) 67–94) on the classical Yamabe-type equation. [ABSTRACT FROM AUTHOR]

A generalized local fractional Riccati differential equation (LFRDE) method, which is a combination of the local fractional Riccati differential equation method and the transformed rational function approach, is presented to obtain the non-differentiable exact traveling wave solutions for n-dimensional fractional partial differential equations. In order to verify the effectiveness of the method, the seventh-order local fractional Sawada–Kotera–Ito equation, the local fractional sine-Gordon equation and the local fractional Kadomtsev–Petviashvili equation in fractal domain are first considered. The obtained results show that the presented method involving fractal special functions, are powerful and effective for obtaining exact solutions of nonlinear fractional partial differential equations in fractal domains. [ABSTRACT FROM AUTHOR]

In this communication we have studied well known physical process of two-dimensional advection–diffusion phenomena. The advection–diffusion equation is time-fractionalized by exploiting Atangana-Baleanu fractional derivative operator. This fractionalization is achieved in the generalized constitutive equation of the mass flux density vector. The fractionalized two-dimensional advection–diffusion equation turns out to be a two-dimensional nonlinear fractional partial differential equation. This partial differential equation is considered under the hypothesis of an initial concentrated loading and Robin type boundary conditions. The analytical expression of the solution is determined for this boundary value problem by employing the integral transforms method, namely, the Laplace transform, sine-Fourier transform and finite sine–cosine Fourier transform. The effects of fractional parameter α on the concentration obtained from the analytical solution, for various parameters of interest, are illustrated graphically with the help of software Mathcad. The graphs illustrate that the memory effects are remarkable for small values of time and ordinary for large values of the time.

In this paper, the invariant subspace method (ISM) is developed to obtain the exact solution of linear and nonlinear time and space fractional mixed partial differential equations involving modified conformable fractional derivative (MCFD). Moreover, the method is successfully extended to the coupled system of modified conformable fractional differential equations. Variety of time-space fractional mixed PDEs are considered and solved to illustrate the established result of ISM. Finally, a comparison between exact solution of fractional PDE in the sense of MCFD and Caputo fractional derivative (Ca-FD) is presented through some examples of time–space fractional mixed partial differential equations.

In the given research, our main goal is to explore new soliton solutions in (2 + 1) -dimensional Heisenberg ferromagnetic spin chain (HFSC) equation. This was achieved via the extended (G ′ / G 2) -expansion method. For this, we first analyse the given (2 + 1) -dimensional HFSC system of a partial differential equation (PDE) that is obtained by separating the equation into real and imaginary parts. The solution of our equation is determined as rapidly convergent sequences effectively computed by using Mathematica software. Furthermore, a special class of nonlinear wave solutions like periodic, straight, decreasing, dark-singular soliton and dark bright singular soliton were obtained in Heisenberg ferromagnetic dynamics. The amplitude of the soliton is determined by giving different values of parameters. With a suitable choice of parameters, 3D and 2D graphical illustrations are reported. The methodology applied is effective for obtaining exact solutions for various fractional partial differential equations in complex medium. The numerical module of the results obtained is studied, with fascinating figures showing the physical significance of the solutions. [ABSTRACT FROM AUTHOR]