Browsing by Author "Barnes, Benedict"

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    Divergence regularization method for solving ill-posed Helmholtz equation
    (June, 2016 ) Barnes, Benedict
    In this work, we introduce Divergence Regularization Method (DRM) for regularizing the Cauchy problem of the Helmholtz equation where the boundary deflection is not equal to zero in Hilbert space H. The DRM incorporates a positive integer scaler which homogenizes inhomogeneous boundary deflection in Cauchy problem of the Helmholtz equation to ensure the existence and uniqueness of solution for the equation. The DRM employs its regualarization term (1 + 2m)em to restore the stability of the regularized Helmholtz equation, and guarantees the uniqueness of solution of Helmholtz equation when it is imposed by Neumann boundary conditions in the upper half-plane. The DRM gives better stability approximation when compared with other methods of regularization for solving Cauchy problem of the Helmholtz equation where the boundary deflection is zero. In the process, we introduce AdaptiveWavelet Spectral Finite Difference (AWSFD) method to obtain the approximated solutions of the regularized Helmholtz equation with regularized Cauchy boundary conditions, regularized Neumann boundary conditions in the upper half-plane, and finally with regularized both Dirichlet and Cauchy boundary conditions where the boundary deflection is equal to zero. The AWSFD method captures the boundary points to obtain approximated solution of Helmholtz equation. This method reduces the Helmholtz equation in two dimensions to one dimension which is then solve spectrally using a suitable wavelet basis. The solutions by AWSFD method confirms the analytic solutions of regularized Helmholtz equation by DRM. The norm of relative error between the analytic solution by DRM and the approximated solution by AWSFD method is minimal. Moreover, we introduce interpolation scheme in the AWSFD method to obtain the approximated solutions of the regularized Helmholtz equation with above boundary conditions.
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    Scattering of kinks in noncanonical sine-Gor Scattering of kinks in noncanonical sine-Gordon Model don Model
    (Turkish Journal of Physics, 2022) Takyi, Ishmael; Barnes, Benedict; Tornyeviadzi, Hoese Michel; Ackora-Prah, Joseph; 0000-0002-1217-0889; 0000-0002-0580-5655; 0000-0001-9488-9610
    In this paper, we numerically study the scattering of kinks in the noncanonical sine-Gordon model using Fourier spectral methods. The model depends on two free parameters, which control the localized inner structure in the energy density and the characteristics of the scattering potential. It has been conjectured that the kink solutions in the noncanonical model possess inner structures in their energy density, and the presence of these yields bound states and resonance structures for some relative velocities between the kink and the antikink. In the numerical study, we observed that the classical kink mass decreases monotonically as the free parameters vary, and yields bion-formations and long-lived oscillations in the scattering of the kink-antikink system. :
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    Vacuum polarization energy of the kinks in the sinh-deformed models
    (Turkish Journal of Physics, 2021) Takyi, Ishmael; Barnes, Benedict; Ackora-Prah, Joseph; 0000-0002-1217-0889; 0000-0002-0580-5655; 0000-0001-9488-9610
    We compute the one-loop quantum corrections to the kink energies of the sinh-deformed 4 and 6 models in one space and one time dimensions. These models are constructed from the well-known polynomial 4 and 6 models by a deformation procedure. We also compute the vacuum polarization energy to the nonpolynomial function = 1 4 (1 − sinh2 .This potential approaches the model in the limit of small values of the scalar function. These energies are extracted from scattering data for fluctuations about the kink solutions. We show that for certain topological sectors with nonequivalent vacua the kink solutions of the sinh-deformed models are destabilized.