Department of Mathematics

Permanent URI for this collectionhttp://localhost:4000/handle/123456789/1920

Browse

Start date: 2020Subject: search.filters.subject.Caputo derivative

Search Results

Now showing 1 - 10 of 10
  • No Thumbnail Available
    Item
    Numerical treatment of multi-term time fractional nonlinear KdV equations with weakly singular solutions
    (Taylor & Francis, 2021-12) Santra, Sudarshan
    The main aim of this work is to construct an efficient recursive numerical technique for solving multi-term time fractional nonlinear KdV equation. The fractional derivatives are defined in Caputo sense. A modified Laplace decomposition method is introduced to approximate the solution. The Adomian polynomials play an important role to execute such a recursive process. In addition, the mathematical importance and some applications of KdV equation are discussed. The approximate solution obtained by the proposed method can be expressed in the form of an infinite convergent series. The experimental evidences demonstrate the effectiveness of the proposed method.
  • No Thumbnail Available
    Item
    Analysis of a finite difference method based on L1 discretization for solving multi-term fractional differential equation involving weak singularity
    (Wiley, 2022-03) Santra, Sudarshan
    In this article, we consider a multi-term fractional initial value problem which has a weak singularity at the initial time . The fractional derivatives are defined in Caputo sense. Due to such singular behavior, an initial layer occurs near which is sharper for small values of γ1 where γ1 is the highest order among all fractional differential operators. In addition, the analytical properties of the solution are provided. The classical L1 scheme is introduced on a uniform mesh to approximate the fractional derivatives. The error analysis is carried out, and it is shown that the numerical solution converges to the exact solution. Further analysis proves that the scheme is of order over the entire region, but it is of order O(τ) on any subdomain away from the origin. τ denotes the mesh parameter. To show the efficiency of the proposed scheme, this method is tested on several model problems, and the results are in agreement with the theoretical findings.
  • No Thumbnail Available
    Item
    An efficient computational approach for the solution of time-space fractional diffusion equation
    (Taylor & Francis, 2022-06) Santra, Sudarshan
    The main aim of this paper is to construct an efficient recursive algorithm to solve a time-space fractional Poisson’s equation which can be treated as a time-space fractional diffusion equation in two dimensions. The fractional derivatives in both time and space are defined in the Caputo sense. A homotopy perturbation method is introduced to approximate the solution, and a comparison is made between the exact and the approximate solutions. In addition, we present a procedure for solving higher-order fractional Poisson’s equations. In this case, the equation is converted to a system of fractional differential equations in which the order of the time derivatives is less than or equal to one. The convergence analysis is carried out, and an apriori bound of the solution is obtained for the present problem. Numerical examples are provided and the experimental evidence proves the effectiveness of the proposed method.
  • No Thumbnail Available
    Item
    Numerical simulation and convergence analysis for Riemann-Liouville fractional initial value problem involving weak singularity
    (Inder Science, 2023-11) Santra, Sudarshan
    The present work considers a Riemann-Liouville fractional initial value problem (IVP) associated with homogeneous initial condition involving a weak singularity near the origin. Due to presence of initial singularity, an initial layer occurs at t = 0. The L1 scheme is introduced on a uniform mesh to approximate the solution. The convergence analysis shows that the present method is more accurate and produces less error compared to some existing methods on any subdomain away from the origin while the proposed method is comparable over the entire region. Numerical examples and comparison results are provided in order to show the effectiveness of the proposed method.
  • No Thumbnail Available
    Item
    Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations
    (Elsevier, 2025-01) Santra, Sudarshan
    An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at . In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.
  • No Thumbnail Available
    Item
    Enhancing accuracy with an adaptive discretization for the non-local integro-partial differential equations involving initial time singularities
    (Elsevier, 2025-08) Santra, Sudarshan
    This work aims to construct an efficient and highly accurate numerical method to address the time singularity at involved in a class of time-fractional parabolic integro-partial differential equations in one and two dimensions. The L2- scheme is used to discretize the time-fractional operator, whereas a modified version of the composite trapezoidal approximation is employed to discretize the Volterra operator in time. Subsequently, it helps to convert the proposed model into a second-order boundary value problem in a semi-discrete form. The multi-dimensional Haar wavelets are then used for grid adaptation and efficient computations for the two-dimensional problem, whereas the standard second-order approximations are employed to approximate the spatial derivatives for the one-dimensional case. The stability analysis is carried out on an adaptive mesh in time. The convergence analysis leads to accurate solution in the space-time domain for the one-dimensional problem having time singularity based on the norm for a suitable choice of the grading parameter. Furthermore, it provides accurate solution for the two-dimensional problem having unbounded time derivative at . The analysis also highlights a higher order accuracy for a sufficiently smooth solution resides in even if the mesh is discretized uniformly. The truncation error estimates for the time-fractional operator, integral operator, and spatial derivatives are presented. In addition, we have examined the impact of various parameters on the robustness and accuracy of the proposed method. Numerous tests are performed on several examples in support of the theoretical analysis. The advancement of the proposed methodology is demonstrated through the application of the time-fractional Fokker-Planck equation and the fractional-order viscoelastic dynamics having weakly singular kernels. It also confirms the superiority of the proposed method compared with existing approaches available in the literature.
  • No Thumbnail Available
    Item
    Physics-informed fractional machine intelligence and space–time wavelet frameworks for non-local integro-partial differential equations involving weak singularities
    (Elsevier, 2026-01) Santra, Sudarshan
    This paper presents a space–time multi-dimensional wavelet framework and a physics-informed fractional machine intelligence (PI-fMI) model to address the weak singularity involved in time-fractional integro-partial differential equations with mixed Volterra–Fredholm operators. Conventional machine learning approaches often struggle with weak initial singularities; however, our proposed approach overcomes this challenge through two complementary strategies in the context of fractional-order integro-differential equations. First, a wavelet-based numerical scheme is employed that utilizes the multi-resolution analysis with the collocation method to compute the wavelet coefficients, ensuring convergence for fractional-order integro-differential problems with sufficiently smooth solutions. Second, we introduce a PI-fMI model for problems that exhibit unbounded temporal derivatives at , which incorporates the discretization for fractional operators, a combination of the repeated quadrature rule, and automatic differentiation to handle integral operators that contain diffusion terms. Theoretical and numerical analyses demonstrate that the proposed approach successfully resolves the initial weak singularities where the traditional Haar wavelets fail to address such issues. Furthermore, the convergence of the PI-fMI model is analyzed for problems with nonlinear source terms, demonstrating its effectiveness under suitable hyperparameter choices. Theoretical findings are validated through extensive numerical experiments on several test problems exhibiting bounded and unbounded temporal derivatives at .
  • No Thumbnail Available
    Item
    Fixed point method for nonlinear fractional differential equations with integral boundary conditions on tetramethyl-butane graph
    (MDPI, 2024-06) Agarwal, Shivi; Mathur, Trilok
    Until now, little investigation has been done to examine the existence and uniqueness of solutions for fractional differential equations on star graphs. In the published articles on the subject, the authors used a star graph with one junction node that has edges with the other nodes, although there are no edges between them. These graph structures do not cover more generic non-star graph structures; they are specific examples. The purpose of this study is to prove the existence and uniqueness of solutions to a new family of fractional boundary value problems on the tetramethylbutane graph that have more than one junction node after presenting a labeling mechanism for graph vertices. The chemical compound tetramethylbutane has a highly symmetrical structure, due to which it has a very high melting point and a short liquid range; in fact, it is the smallest saturated acyclic hydrocarbon that appears as a solid at a room temperature of 25 °C. With vertices designated by 0 or 1, we propose a fractional-order differential equation on each edge of tetramethylbutane graph. Employing the fixed-point theorems of Schaefer and Banach, we demonstrate the existence and uniqueness of solutions for the suggested fractional differential equation satisfying the integral boundary conditions. In addition, we examine the stability of the system. Lastly, we present examples that illustrate our findings.
  • No Thumbnail Available
    Item
    Fractional-order crime propagation model with non-linear transmission rate
    (Elsevier, 2023-04) Agarwal, Shivi; Mathur, Trilok
    Various studies present different mathematical models of ordinary and fractional differential equations to reduce delinquent behavior and encourage prosocial growth. However, these models do not consider the non-linear transmission rate, which depicts reality better than the linear transmission rate, as the relationship between non-criminals and criminals is not linear. In light of this, a novel fractional-order mathematical crime propagation model with a non-linear Beddington–DeAngelis transmission rate is proposed that divides the entire population into three clusters. The present study also compares the crime transmission models for various transmission rates, followed by an analytical investigation. The model shows two equilibrium points (criminal-free and crime-persistence equilibrium). The criminal-free equilibrium is locally and globally asymptotically stable when the criminal generation number is less than one. The crime-persistence equilibrium point does not appear until the criminal generation number exceeds one. In addition, this research investigates the incidence of transcritical bifurcation at the criminal-free equilibrium point. Furthermore, numerical simulations are performed to demonstrate the analytical results. In summary, the finding of this research suggests that as the order of derivative increases, the population approaches equilibrium more swiftly, and criminals decline with time for the different order of derivative.
  • No Thumbnail Available
    Item
    Underlying dynamics of crime transmission with memory
    (Elsevier, 2021-05) Agarwal, Shivi; Mathur, Trilok
    Various studies suggest different mathematical models of integer order differential equations predict crime. But these models do not inherit non-local property, which depicts behavior changes due to contact with criminals for a long period. To overcome this, a fractional-order mathematical model of crime transmission is proposed in this study. The proposed model considers the previous effects of the input while predicting the crime growth rate. A mathematical model of crime transmission inherited with memory property is proposed in this study to analyze crime congestion. Abstract compartmental parameters of fractional crime transmission equation, which illustrates various stages of criminal activity, were employed to analyze crime contagion in the society. The present study demonstrates the progression of the flow of population by classifying into three systems based on involvement in crime and imprisonment by considering the criminal history of an individual. Subsequently, the equilibria of the three-dimensional fractional crime transmission model are evaluated using phase-plane analysis. The Lyapunov function is employed to determine threshold conditions to achieve a crime-free society.