Fractional Partial Differential Equations Research Articles

A novel Adomian natural decomposition method with convergence analysis of nonlinear time-fractional differential equations

ABSTRACT The objective of the current article is to use the Banach fixed point theorem to present proofs for the existence and uniqueness theorems for a nonlinear time-fractional differential equation, namely, the linear and nonlinear, homogeneous and nonhomogeneous time-fractional diffusion equations. In addition, we explore exact solutions to nonlinear time fractional partial differential equations using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). To demonstrate the effectiveness of the suggested scheme, three examples are provided along with graphs and numerical tables to support our work. The outcomes demonstrate how effortless and effective the ANDM is for addressing fractional partial differential equations in multi-dimensional spaces, some of which we examined in this study as special cases. AMS Classifications 34A08, 65P99, 49J15, 35R11, 26A33, 74G10.

Non-autonomous fractional nonlocal evolution equations with superlinear growth nonlinearities

We carry out an analysis of the existence of solutions for a class of nonlinear fractional partial differential equations of parabolic type with nonlocal initial conditions. Sufficient conditions for the solvability of the desired problem are presented by transforming it into an abstract non-autonomous fractional evolution equation, and constructing two families of solution operators based on the Mittag-Leffler function, the Mainardi Wright-type function and the analytic semigroup generated by the closed densely defined operator −A(⋅). The discussions are based on the fractional power theory as well as the Banach fixed point theorem in the interpolation space Xpυ (0⩽υ<1, 1<p<∞).

A Robust Numerical Simulation of a Fractional Black–Scholes Equation for Pricing American Options

After the discovery of fractal structures of financial markets, fractional partial differential equations (fPDEs) became very popular in studying dynamics of financial markets. Available research results involves two key modelling aspects; firstly, derivation of tractable asset pricing models, those that closely reflects the actual dynamics of financial markets. Secondly, the development of robust numerical solution methods. Often times, most the effective models are of a nonlinear nature, and as such, reliable analytical solution methods are seldomly available. On the other hand, the accurate value of American options strongly lies on the unknown free boundaries associated with these types of derivative contracts. The free boundaries emanates from the flexibility of the early exercise features with American options. To the best of our knowledge, the approach of pricing American options under the fractional calculus framework has not been extensively explored in literature, and an obvious wider research gap still exist on the design of robust solution methods for pricing American option problems formulated under the fractional calculus framework. Therefore, this paper serve to propose a robust numerical scheme for solving time-fractional Black–Scholes PDEs for pricing American put option problems. The proposed scheme is based on the front-fixing algorithm, under which the early exercise boundaries are transformed into fixed boundaries, allowing for a simultaneous computation of optimal exercise boundaries and their corresponding fair premiums. Results herein indicate that, the proposed numerical scheme is consistent, stable, convergent with order O(h2,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(h^2,k)$$\\end{document}, and also does guarantee positivity of solutions under all possible market conditions.

An Efficient Numerical Solution of a Multi-Dimensional Two-Term Fractional Order PDE via a Hybrid Methodology: The Caputo–Lucas–Fibonacci Approach with Strang Splitting

An Efficient Numerical Solution of a Multi-Dimensional Two-Term Fractional Order PDE via a Hybrid Methodology: The Caputo–Lucas–Fibonacci Approach with Strang Splitting

Global existence and stability results for a time-fractional diffusion equation with variable exponents

This paper aims to study the existence and stability results concerning a fractional partial differential equation with variable exponent source functions. The local existence result for α∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1)$$\\end{document} is established with the help of the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-resolvent kernel and the Schauder-fixed point theorem. The non-continuation theorem is proved by the fixed point technique and accordingly the global existence of solution is achieved. The uniqueness of the solution is obtained using the contraction principle and the stability results are discussed by means of Ulam-Hyers and generalized Ulam-Hyers-Rassias stability concepts via the Picard operator. Examples are provided to illustrate the results.

Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs

This paper introduces gradient-based adaptive neural networks to solve local fractional elliptic partial differential equations. The impact of physics-informed neural networks helps to approximate elliptic partial differential equations governed by the physical process. The proposed technique employs learning the behaviour of complex systems based on input-output data, and automatic differentiation ensures accurate computation of gradient. The method computes the singularity-embedded local fractional partial derivative model on a Hausdorff metric, which otherwise halts the computation by available approximating numerical methods. This is possible because the new network is capable of updating the weight associated with loss terms depending on the solution domain and requirement of solution behaviour. The semi-positive definite character of the neural tangent kernel achieves the convergence of gradient-based adaptive neural networks. The importance of hyperparameters, namely the number of neurons and the learning rate, is shown by considering a stationary anomalous diffusion-convection model on a rectangular domain. The proposed method showcases the network’s ability to approximate solutions of various local fractional elliptic partial differential equations with varying fractal parameters.

Shape and location recovery of laser excitation sources in photoacoustic imaging using topological gradient optimization

This study focuses on source term inversion in fractional partial differential equations, specifically applied to photoacoustic imaging. This work contributes to advancing imaging techniques and provides practical insights for medical diagnostics and materials characterization. Our aim in this paper is to develop an accurate method for recovering the location and the shape of a laser excitation source from partial boundary data. To achieve this, we reformulate our inverse problem as an optimization challenge. We utilize topological sensitivity analysis to establish an asymptotic expansion of a relevant shape function. These theoretical findings form the basis for a rapid and precise detection algorithm. Additionally, we present several numerical experiments that demonstrate the effectiveness and accuracy of our proposed approach.

An adaptive algorithm for numerically solving fractional partial differential equations using Hermite wavelet artificial neural networks

This study aims to develop a new strategy for solving partial differential equations with fractional derivatives (FPDEs) using artificial neural networks (ANNs). Numerical solutions to FPDEs are obtained through the Hermite wavelet neural network (HWNN) model. The Caputo fractional derivative is consistently applied throughout the research to address fractional-order partial differential problems. To enhance computational efficiency and expand the input pattern, the hidden layer is removed. A neural network (NN) model featuring a feed-forward architecture and error-back propagation without supervision is employed to optimize network parameters and minimize errors. Numerical illustrations are presented to demonstrate the effectiveness of this approach in preserving computational efficiency while solving FPDEs.

Nonlinear dynamic wave characteristics of optical soliton solutions in ion-acoustic wave

Traveling wave solutions are utilized to depict reaction-diffusion and analyze electrical signal transmission and propagation in space-time nonlinear fractional order partial differential equations like the space-time fractional Telegraph and Kolmogorov-Petrovsky Piskunov equations, and that are used in physical science to model combustion, biological research to model nerve impulse propagation, chemical dynamics to model concentration in order wave propagation, and plasma to model the progression of a set of duffing oscillators. In this study, the new generalized (G′/G)−expansion technique was employed to construct some novel and more universal closed-form traveling wave solutions in the sense of conformable derivatives which explain the above-stated phenomena properly. By utilizing complex fractional transformation, the ordinary differential equations are generated from fractional order differential equations. The recommended technique allowed us to produce some dynamical wave patterns of kink, single soliton, compacton, periodic shape, multiple periodic waves, anti-kink, and other structures are developed, which are shown using 3D plots and contour plots to illustrate the physical layout clearly. The traveling waveform responses can be defined in terms of functions based on trigonometry, hyperbolic operations, and rational functions and that are quick, flexible, and simple to reproduce. Furthermore, the obtained closed-form solutions for nonlinear fractional evolution equations make stability analysis and accuracy comparison amongst numerical solvers easier, which asserts that the new generalized (G′/G)−expansion technique is one of the most proficient and effective approaches.

Abundant Closed-Form Soliton Solutions to the Fractional Stochastic Kraenkel–Manna–Merle System with Bifurcation, Chaotic, Sensitivity, and Modulation Instability Analysis

An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional stochastic Kraenkel–Manna–Merle system with some important analyses, such as bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability. This fractional system renders a substantial impact on signal transmission, information systems, control theory, condensed matter physics, dynamics of chemical reactions, optical fiber communication, electromagnetism, image analysis, species coexistence, speech recognition, financial market behavior, etc. The Sardar sub-equation approach was implemented to generate several genuine innovative closed-form soliton solutions. Additionally, phase portraiture of bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability were employed to monitor the qualitative characteristics of the dynamical system. A certain number of the accumulated outcomes were graphed, including singular shape, kink-shaped, soliton-shaped, and dark kink-shaped soliton in terms of 3D and contour plots to better understand the physical mechanisms of fractional system. The results show that the proposed methodology with analysis in comparison with the other methods is very structured, simple, and extremely successful in analyzing the behavior of nonlinear evolution equations in the field of fractional PDEs. Assessments from this study can be utilized to provide theoretical advice for improving the fidelity and efficiency of soliton dissemination.

A new approach of B-spline wavelets to solve fractional differential equations

This paper presents a groundbreaking method for solving the multi-order fractional differential (M-OFD), both linear and nonlinear, as well as fractional partial differential equations (FPDE)s. This approach involves constructing an operational matrix of fractional derivatives using linear B-spline (LB-S) wavelet functions with perfect subtlety. The new method has two crucial features. Firstly, it simplifies the problem by converting it into a set of algebraic equations, which is a significant advantage. This makes the method highly accurate and reliable. Secondly, it uses thresholding to dramatically reduce the computational workload in linear problems. This leads to lightning-fast and highly efficient problem-solving. The newly developed scheme underwent a thorough examination of its error estimates and convergence, revealing remarkable results in terms of accuracy and efficiency. The analysis provides a comprehensive understanding of the scheme’s performance, highlighting its potential as a dependable and effective method. Based on the findings, it is evident that the proposed method not only delivers exceptional precision but also operates with remarkable efficiency.

Optical solutions for perturbed conformable Fokas–Lenells equation via Kudryashov auxiliary equation method

This paper is dedicated to the study of optical soliton solutions for the perturbed Fokas–Lenells equation with conformable derivative using the Kudryashov auxiliary equation method. The studied optical solutions include a class of categories, comprising dark, mixed dark-bright, multi bell-shaped, bell-shaped, and wave optical solutions. Furthermore, we analyzed the magnitude of the perturbed conformable Fokas–Lenells equation by investigating the impact of the conformable parameter and the effect of the time parameter on the novel optical solutions. It can be claimed that the current optical soliton solutions are novel and have not existed in the literature. The results obtained illustrate that the proposed method is robust, efficient, and readily applicable for constructing new solutions to a wide range of nonlinear fractional partial differential equations. The results of this study are expected to shed light on the field of soliton theory in nonlinear optics and mathematical physics.

Galerkin-finite difference method for fractional parabolic partial differential equations

The fractional form of the classical diffusion equation embodies the super-diffusive and sub-diffusive characteristics of any flow, depending on the fractional order. This study aims to approximate the solution of parabolic partial differential equations of fractional order in time and space. For this, firstly, we briefly discussed on some existing methods to solve Partial Differential Equations (PDEs) and fractional Differential Equations (DEs), then introduce a combined technique such as the Galerkin weighted residual method for the space fractional term with modified Bernoulli polynomials as basis functions, and the finite difference approximation for the time fractional term, respectively. The mathematical formulation of the proposed method is explained elaborately. Then we describe the order of convergence for the time fractional term only, as the convergence of the Galerkin method is obvious. We impose this technique on the fractional Black-Scholes model subsequently. Finally, we experimented with our proposed technique on some numerical problems. All the results are depicted in both tabular and 3D visualizations as well. We compare our results with the available methods in the literature, and our accuracy is considerable. To summarize: •The paper introduces an approach that integrates the Galerkin weighted residual method with modified Bernoulli polynomials to handle space fractional terms, alongside employing a finite difference approximation for time fractional terms.•The convergence analysis is focused.•The technique is implemented on the fractional Black-Scholes model and other numerical problems, with outcomes depicted through tables and 3D visualizations.

Decomposing Method for Space-Time Fractional Order PDEs

Numerical solutions to integer-order differential equations frequently employ decomposition as one of many methods. This paper provides a generalization of integer-order differential equations into fractional order, which is more general, as well as generalizing the decomposing method into the fractional case and then applying it to solve differential equations of fractional order in both space and time. A fractional calculus's theorems and properties to generalize both differential equations and the decomposition method have been utilized. This method to solve different fractional time and space partial differential equations, demonstrating its ability and applicability to various problems have been applied. The method to various cases to highlight its strengths has been used. The graphs and tables of results obtained using Matlab programs have been presented and discussed.

An innovative approach to approximating solutions of fractional partial differential equations

The RPS-M (residual power series method) is a valuable technique for solving F-PDEs (fractional partial differential equations). However, the derivative of the residual function to obtain the coefficients of the series is required in RPS-M. This makes the application of the classical RPS-M limited to a certain extent due to the complexity of the derivation of the residual function for higher iterations. To overcome this obstacle, in this study, we present a simplified version of this approach with the help of Laplace transform that requires less computation and offers higher accuracy. This modified method does not require derivation as well as limit of the residual function to estimate the unknown coefficients of the series solution. To demonstrate its effectiveness, we apply the proposed method to nonlinear F-PDEs to obtain their semi-analytical solution. The obtained solutions exhibit excellent agreement when compared to results obtained using other established approaches. We have also provided the convergence analysis of the obtained solution. Furthermore, by comparing the outcomes for various values of the non-integer order σ, we observe that as the value approaches an integer order, the solution converges towards the exact solution.

A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

This work presents a thorough analysis of soliton wave phenomena in the (3+1)-dimensional Fractional Calogero-Bogoyavlenskii-Schiff equation (FCBSE) with Caputo’s derivatives through the use of a novel analytical technique known as the modified Extended Direct Algebraic Method (mEDAM). By converting nonlinear Fractional Partial Differential equations (FPDE) into integer-order Nonlinear Ordinary Differential equations (NODE), and then using closed-form series solutions to translate the NODE into an algebraic system of equations, this method allows us to derive families of soliton solutions, which include kink waves, lump waves, breather waves, and periodic waves, exposing new insights into the behavior and distinctive features of soliton waves in the FCBSE. By including contour and 3D graphics, the behaviors of a few selected soliton solutions are well depicted, showcasing their amplitude, shape, and propagation characteristics. The results enhance our understanding of the FCBSE and show that the mEDAM is a valuable tool for studying soliton wave phenomena. This work creates new opportunities for studying wave phenomena in more intricately constructed nonlinear FPDEs (NFPDEs).

ON THE PERIODIC SOLITON SOLUTIONS FOR FRACTIONAL SCHRÖDINGER EQUATIONS

In this research, we use a novel version of the Extended Direct Algebraic Method (EDAM) namely generalized EDAM (gEDAM) to investigate periodic soliton solutions for nonlinear systems of fractional Schrödinger equations (FSEs) with conformable fractional derivatives. The FSEs, which is the fractional abstraction of the Schrödinger equation, grasp notable relevance in quantum mechanics. The proposed gEDAM technique entails creating nonlinear ordinary differential equations via a fractional complex transformation, which are then solved to acquire soliton solutions. Several 3D and contour graphs of the soliton solutions reveal periodicity in the wave profiles that offer crucial perspectives into the behavior of the system. The work sheds light on the dynamics of FSEs by displaying numerous families of periodic soliton solutions and their intricate relationships. These results hold significance not only for comprehending the dynamics of FSEs but also for nonlinear fractional partial differential equation applications.

A new identification of Lagrange multipliers to study solutions of nonlinear Caputo–Fabrizio fractional problems

This research aims to provide a new identification of Lagrange multipliers using a novel hybrid method based on the Khalouta transform method and the variational iteration method and to study the solutions of nonlinear fractional problems. This method is called Khalouta variational iteration method (KHVIM). The fractional derivative is considered in Caputo–Fabrizio sense. The uniqueness and convergence results are investigated using Banach fixed point theorem. The results are obtained in the form of successive approximations corresponding to the proposed problem. To demonstrate the accuracy and effectiveness of the proposed method, three different nonlinear fractional partial differential equations are provided. Approximate solutions of the fractional equations were obtained. These solutions quickly converged to exact solutions with lower computational cost. Furthermore, the method used in this study is more generalized and allows our results to be more extensive and cover several new and existing fractional problems in the literature.

Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations

In this study, we use the Khater Method (KM) as an efficient analytical tool to solve (3+1)-dimensional fractional extended shallow water wave equations (FESWWEs) with conformable derivatives. The KM transforms fractional partial differential equations to ordinary differential equations (ODEs) via strategic variable transformation. Then, series-form solutions to these ODEs are proposed, which turn them into nonlinear algebraic systems. The solution to this set of algebraic equations yields shock travelling wave solutions expressed in hyperbolic, trigonometric, exponential, and rational functions. The study’s findings are corroborated by 2D, 3D, and contour graphs that show the changing patterns of the detected shock travelling waves. These findings have important significance for the discipline, offering vital insights into the intricate dynamics of FESWWEs. The effectiveness of KM is demonstrated by its capacity to produce varied solutions and contribute to a thorough knowledge of such complex phenomena.

Lie Symmetry Analysis, Power Series Solutions and Conservation Laws of (2+1)-Dimensional Time Fractional Modified Bogoyavlenskii–Schiff Equations

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations, which is an important model in physics. The one-dimensional optimal system composed by the obtained Lie symmetries is utilized to reduce the system of (2+1)-dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to the system of (1+1)-dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative. Then the power series method is applied to derive explicit power series solutions for the reduced system. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.