Advantages of the Differential Equations for Solving Problems in Mathematical Physics with Symbolic Computation

Abdoon, Mohamed A.; Hasan, Faeza Lafta

doi:10.18280/mmep.090133

Cited by 17 publications

(12 citation statements)

References 21 publications

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“…One of the primary challenges lies in the computational domain. Future work could address these challenges by exploring the following avenues: The development of more advanced numerical algorithms that can reduce approximation errors and enhance the efficiency of simulations, making them more suitable for high-dimensional systems and real-time analysis, In-depth exploration of the interactions between memristive properties and fractional-order dynamics, potentially through the lens of other non-integer derivatives or through the adoption of alternative memristor models, Investigating the impact of system parameters on the robustness and sensitivity of the chaotic behaviours observed to better control and harness these dynamics for practical applications see [1,3,4,[26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Analysis and Applications of the Chaotic Hyperbolic Memristor Model with Fractional Order Derivative

Ahmed

2024

Eur. J. Pure Appl. Math.

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This work presents a new five-dimensional fractional-order chaotic system that includes a feedback memristor. A Lorenz-Stenflo-based fractional-order chaotic system with five dimensions that contains feedback memristor dynamics is used to do this. This work focuses on chaotic models through the ABC fractional derivative, a concept we rigorously examine. We designed and usedsophisticated numerical algorithms to model fractional-order dynamics with the utmost precision to understand this system’s complex characteristics. These numerical approaches excel at non-integer order differentiation, which standard numerical methods struggle with. Our research uses fractional calculus to increase the system’s complexity and robustness, revealing its hyperchaotic nature. We demonstrate that the system is chaotic and stable over fractional orders using eigenvalues, Lyapunov exponents, Kaplan-Yorke dimensions, maximal exponents, phase portraits, and equilibrium points. Our conclusions are strengthened by using these numerical systems, intended for this work, and analytical tools. We improve our understanding of fractional-order chaotic systems with our research. It illuminates how to improve electrical integrations and create more sophisticated nonlinear dynamical systems. This work establishes the approach and insights for future research, guiding the development of new systems with enhanced dynamical features

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Section: Introductionmentioning

confidence: 99%

Analysis and Applications of the Chaotic Hyperbolic Memristor Model with Fractional Order Derivative

Ahmed

2024

Eur. J. Pure Appl. Math.

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“…Differential equations play a crucial role in modelling a diverse range of physical [1], biological [2], and engineering phenomena [3,4]. Achieving an optimal balance between accuracy and computational efficiency in solving these equations has been a longstanding challenge.…”

Section: Introductionmentioning

confidence: 99%

Physics Informed Deep Learning Approach for Differential Equation

2024

nano-ntp

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This study presents a Physics-Informed Neural Network (PINN) approach for solving differential equations, providing a versatile and data-driven alternative to traditional methods. Focusing on the comparison between exact, analytic, and neural network solutions, we investigate the effectiveness of PINNs in capturing complex dynamics across diverse applications. The comparison is illustrated through carefully crafted graphs, highlighting the accuracy and efficiency of the PINN methodology. By eliminating the need for explicit analytical solutions, PINNs offer a flexible framework for addressing a wide range of differential equations, showcasing their potential to revolutionize problem-solving in various scientific and engineering domains. This research contributes valuable insights into the capabilities of PINNs and their role in advancing computational methodologies for differential equations.

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“…Fractional calculus has extensive use across various industries like engineering, physics, finance, and signal processing. It offers an advanced approach to comprehend phenomena like as anomalous diffusion, fractal time series, and, viscoelasticity, all of which exhibit fractional or fractal characteristics [6][7][8][9][10][11][12][13][14]. In addition, fractional calculus provides a potent mathematical tool for modeling and examining complex systems that cannot be entirely elucidated by employing the standard methods of integer-order calculus [15].…”

Section: Introductionmentioning

confidence: 99%

Non-Standard Finite Difference and Vieta-Lucas Orthogonal Polynomials for the Multi-Space Fractional-Order Coupled Korteweg-de Vries Equation

Saad,

Srivastava

2024

Symmetry

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This paper focuses on examining numerical solutions for fractional-order models within the context of the coupled multi-space Korteweg-de Vries problem (CMSKDV). Different types of kernels, including Liouville-Caputo fractional derivative, as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, are utilized in the examination. For this purpose, the nonstandard finite difference method and spectral collocation method with the properties of the Shifted Vieta-Lucas orthogonal polynomials are employed for converting these models into a system of algebraic equations. The Newton-Raphson technique is then applied to solve these algebraic equations. Since there is no exact solution for non-integer order, we use the absolute two-step error to verify the accuracy of the proposed numerical results.

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Advantages of the Differential Equations for Solving Problems in Mathematical Physics with Symbolic Computation

Cited by 17 publications

References 21 publications

Analysis and Applications of the Chaotic Hyperbolic Memristor Model with Fractional Order Derivative

Analysis and Applications of the Chaotic Hyperbolic Memristor Model with Fractional Order Derivative

Physics Informed Deep Learning Approach for Differential Equation

Non-Standard Finite Difference and Vieta-Lucas Orthogonal Polynomials for the Multi-Space Fractional-Order Coupled Korteweg-de Vries Equation

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