Imaging Noise Suppression: Fourth-Order Partial Differential Equations and Travelling Wave Solutions

Jamal, Sameerah

doi:10.3390/math8112019

Cited by 4 publications

(2 citation statements)

References 20 publications

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“…The classical heat equation has appeared in many studies [26,27], and a fourth-order nonlocal heat model was recently explored in [28]. Indeed, we showcase here that the heat equation is of the most useful of PDEs.…”

Section: The Heat Equationmentioning

confidence: 64%

Fractional Pricing Models: Transformations to a Heat Equation and Lie Symmetries

Champala,

Jamal,

Khan

2023

Fractal Fract

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The study of fractional partial differential equations is often plagued with complicated models and solution processes. In this paper, we tackle how to simplify a specific parabolic model to facilitate its analysis and solution process. That is, we investigate a general time-fractional pricing equation, and propose new transformations to reduce the underlying model to a different but equivalent problem that is less challenging. Our procedure leads to a conversion of the model to a fractional 1 + 1 heat transfer equation, and more importantly, all the transformations are invertible. A significant result which emerges is that we prove such transformations yield solutions under the Riemann–Liouville and Caputo derivatives. Furthermore, Lie point symmetries are necessary to construct solutions to the model that incorporate the behaviour of the underlying financial assets. In addition, various graphical explorations exemplify our results.

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Section: The Heat Equationmentioning

confidence: 64%

Fractional Pricing Models: Transformations to a Heat Equation and Lie Symmetries

Champala,

Jamal,

Khan

2023

Fractal Fract

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“…The HTE admits a fundamental solution in terms of the Gaussian function and many studies of parabolic PDEs involve the HTE [36,37]. The HTE offers many benefits, but primarily, we focus on its symmetry properties.…”

Section: The Heat Transfer Equationmentioning

confidence: 99%

Lie Symmetries and the Invariant Solutions of the Fractional Black–Scholes Equation under Time-Dependent Parameters

Jamal,

Champala,

Khan

2024

Fractal Fract

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In this paper, we consider the time-fractional Black–Scholes model with deterministic, time-varying coefficients. These time parametric constituents produce a model with greater flexibility that may capture empirical results from financial markets and their time-series datasets. We make use of transformations to reduce the underlying model to the classical heat transfer equation. We show that this transformation procedure is possible for a specific risk-free interest rate and volatility of stock function. Furthermore, we reverse these transformations and apply one-dimensional optimal subalgebras of the infinitesimal symmetry generators to establish invariant solutions.

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On the conservation laws, Lie symmetry analysis and power series solutions of a class of third-order polynomial evolution equations

2023

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In the present paper, we consider a special class of third-order polynomial evolutionary equations. These equations, via Lie theory admit the same one-parameter point transformations which leave the equations invariant. Reductions with these invariant functions lead to highly nonlinear third-order ordinary differential equations. We use a power series to establish interesting solutions to the reduced equations, whereby recurrence relations occur and convergence of the series may be tested. Finally, the conserved vectors of the class are constructed.

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Imaging Noise Suppression: Fourth-Order Partial Differential Equations and Travelling Wave Solutions

Cited by 4 publications

References 20 publications

Fractional Pricing Models: Transformations to a Heat Equation and Lie Symmetries

Fractional Pricing Models: Transformations to a Heat Equation and Lie Symmetries

Lie Symmetries and the Invariant Solutions of the Fractional Black–Scholes Equation under Time-Dependent Parameters

On the conservation laws, Lie symmetry analysis and power series solutions of a class of third-order polynomial evolution equations

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