Phase-lag mixed integral equation of a generalized symmetric potential kernel and its physical meanings in (3+1) dimensions Azhar Rashad Jan Heliyon, 2025 Currently, there has been no research conducted on the impact of time on the structure of the overall nucleus, nor has there been any investigation into the influence of time delay on the three-dimensional integral equation, considering the temporal changes. This research is being studied to understand its meaning. In order to analyze a phase-lag mixed integral equation (P-LMIE) in dimensions (3+1), in L 2 ( Ω ) × C [ 0 , T ] , T < 1 , where Ω = { ( x , y , z ) ∈ Ω : x 2 + y 2 ≤ a , z = 0 } is the position domain of integration and T is the time. Some specific assumptions were established. The position kernel was imposed, according to Hooke's law, as a generalized potential function in L 2 ( Ω ) . By applying the properties of fractional calculus, it is possible to get an integro-differential Fredholm-Volterra integral equation ( Io-DF-VIE ). The kernel employs the generalized Weber-Sonien integral formula by utilizing polar coordinates. Moreover, the separation approach is utilized to convert the MIE into m-harmonic Fredholm integral equations ( FIEs ) with kernels expressed in the Weber-Sonien integral forms and coefficients involving both temporal and fractional components. The degenerate method is employed to deduce the linear algebraic system ( LAS ). In addition, our endeavor yielded novel and distinct instances. In addition, Maple 2018 and mathematical programming are utilized to calculate numerical values for various coefficients related to the Weber-Sonien integral and its harmonic degree.
Analytical and Numerical Treatment of an Integro Partial Differential Equation in Position and Time with an Anomalous Position Kernel Azhar Rashad Jan Contemporary Mathematics Singapore, 2025 This work presents a second-order Integro-Partial Differential Equation (Io-PDE) with respect to time and space, incorporating a generalized anomalous spatial kernel k(|x−y|), −1 ≤ x, y ≤ 1. From this general kernel, one can derive many special kernels, such as the logarithmic and Carleman types, the Cauchy kernel, and the strongly anomalous kernel. Other special cases can also be derived from the proposed generalized kernel. A delayed-phase formulation is also extracted as a specific instance. By imposing initial conditions, the Io-PDE is reformulated as a Mixed Integral Equation (MIE) defined in both space and time domains. We establish the uniqueness and existence of a solution and demonstrate the convergence properties. A separation of variables approach is then applied, leading to a System of Fredholm Integral Equations (SFIEs) characterized by singular spatial kernels and time-dependent coefficients. The Toeplitz Matrix Method (TMM), known for its robustness in handling anomalous equations, is utilized to numerically solve these SFIEs. This method simplifies complex anomalous integrals into standard numerical forms. Numerical experiments are conducted using logarithmic and Carleman kernels, and associated error metrics are evaluated.
On a Model for Solving Mixed Fractional Integro Differential Equation Azhar Rashad Jan Contemporary Mathematics Singapore, 2024 In this work, the mixed fractional integro differential equation (MfrIo-DE) of the second kind, under certain condition is considered, in the space L2(−1, 1)× C [0, T]; T < 1 T is the time. The position kernel k (|x−y|) of IE has a singularity. After integrating and using the properties of fractional integral, we have a MIE in position and time, where the kernel of position takes the singular form k (|x−y|), and the kernel of time takes the singular Abel form (t −τ)α−1 , 0 < α < 1. Then, using separation of variable method, under certain substitution, we obtain FIE in position, with variable fractional coefficients in time. Using the Toeplitz matrix method (TMM), we have a nonlinear algebraic system (NAS). Moreover, numerical results are obtained and discussed, especially when 0 < α < 1. Also, the solutions of the mixed equation are considered when α = 0, α = 1. Finally, the error estimate, in each case, is computed.
A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a Quadrature Nystrom Method A. R. Jan, M. A. Abdou, M. Basseem Fractal and Fractional, 2023 In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space L2Ω×C0,T, T<1. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS).
Solution of nonlinear mixed integral equation via collocation method basing on orthogonal polynomials A.R. Jan Heliyon, 2022 of the first, second and third kind with continuous kernels in position and time, are considered. In addition, by considering different times of the proposed method and using Mable 18, many numerical results are computed. Moreover, the error estimate, in each case, is calculated.
An Asymptotic Model for Solving Mixed Integral Equation in Position and Time A. R. Jan Journal of Mathematics, 2022 In this paper, we considered a mixed integral equation (MIE) of the second kind in the space L2[−b, b] × C[0, T], T < 1. The kernel of position has a singularity and takes some different famous forms, while the kernels of time are positive and continuous. Using an asymptotic method of separating the variables, we have a Fredholm integral equation (FIE) in position with variable parameters in time. Then, using the Toeplitz matrix method (TMM), we obtain a linear algebraic system (LAS) that can be solved numerically. Some applications with the aid of the maple 18 program are discussed when the kernel takes Coleman function, Cauchy kernel, Hilbert kernel, and a generalized logarithmic function. Also the error estimate, in each case, is computed.
