# Iranian Journal of Numerical Analysis and Optimization Volume:13 Issue: 4, Autumn 2023

• تاریخ انتشار: 1402/09/10
• تعداد عناوین: 10
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• Z. Sarvari * Pages 578-603
In this paper, a high order accuracy method is developed for finding the approximate solution of two-point boundary value problems. The present approach is based on a special algorithm, taken from Pascal’s triangle, for obtaining a generalized form of the parametric splines of degree (2k + 1), k = 1, 2, . . . , which has a lower computational cost and gives the better ap-proximation. Some appropriate band matrices are used to obtain a matrix form for this algorithm.The approximate solution converges to the exact solution of order O(h4k ), where k is a quantity related to the degree of parametric splines and the number of matrix bands that are applied in this paper. Some examples are given to illustrate the applicability of the method, and we compare the computed results with other existing known methods. It isobserved that our approach produced better results.
Keywords: Boundary value problems, Parametric spline, Band matrices, Pascal’s triangle
• G. Ajileye *, T. Oyedepo, L. Adiku, J. Sabo Pages 604-626
In this paper, the standard collocation approach is used to solve multi-order fractional integro-differential equations using Caputo sense. We obtain the integral form of the problem and transform it into a system of linear alge-braic equations using standard collocation points. The algebraic equations are then solved using the matrix inversion method. By substituting the algebraic equation solutions into the approximate solution, the numerical result is obtained. We establish the method’s uniqueness as well as the convergence of the method. Numerical examples show that the developed method is efficient in problem-solving and competes favorably with the existing method.
Keywords: Integro-differential equations, Collocation method, Fredholm-Volterra equations, Multi- order
• N.T. Negero * Pages 627-645
A singularly perturbed time delay parabolic problem with two small pa-rameters is considered. The paper develops a finite difference scheme that is exponentially fitted on a uniform mesh in the spatial direction and uses the implicit-Euler method to discretize the time derivative in the temporal direction in order to obtain a better numerical approximation to the solu-tions of this class of problems. We establish the parameter-uniform error estimate and discuss the stability of the suggested approach. In order to demonstrate the improvement in terms of accuracy, numerical results are also shown to validate the theoretical conclusions and are contrasted with the current hybrid scheme.
Keywords: Singular perturbation, Two parameters parabolic convection-diffusion problem, Time delay, Fitted operator scheme, the Error estimate
• N. Hamidat, S.M. Bahri, N. Abbassa * Pages 646-671
In this work, we present a new mathematical model for the spread of hepatitis C disease in two populations: human population and medical equipment population. Then, we apply the Jacobi wavelets method com-bined with the decoupling and quasi-linearization technique to solve this set of nonlinear differential equations for numerical simulation.
Keywords: Hepatitis C, sterilization, Jacobi wavelets, Operational matrix of derivative, simulation
• N. Mollahasani * Pages 672-694
In this paper, a new orthogonal system of nonpolynomial basis functions is introduced and used to solve a class of time-fractional partial differential equations that have nonsmooth solutions. In fact, unlike polynomial bases, such basis functions have singularity and are constructed with a fractional variable change on Hahn polynomials. This feature leads to obtaining more accurate spectral approximations than polynomial bases. The introduced method is a spectral method that uses the operational matrix of fractional order integral of fractional-order shifted Hahn functions and finally convertsthe equation into a matrix equation system. In the introduced method, no collocation method has been used, and initial and boundary conditions are applied during the execution of the method. Error and convergence analysis of the numerical method has been investigated in a Sobolev space. Finally, some numerical experiments are considered in the form of tables and figures to demonstrate the accuracy and capability of the proposed method.
Keywords: Fractional-order shifted Hahn functions, Fractional-time partial differential equations, Spectral method, Error analysis, Convergence Analysis
• M. Askari * Pages 695-710
The aim of this article is to present a new method based on Lucas poly-nomials and residual error function for a numerical solution of fractional Bagley–Torvik equations. Here, the approximate solution is expanded as a linear combination of Lucas polynomials, and by using the collocation method, the original problem is reduced to a system of linear equations. So, the approximate solution to the problem could be found by solving this system. Then, by using the residual error function and approximating the error function by utilizing the same approach, we achieve more accurate results. In addition, the convergence analysis of the method is investi-gated. Numerical examples demonstrate the validity and applicability of the method.
Keywords: Fractional Bagley–Torvik equation, Caputo derivative, Lucas polynomials, Residual error function, Convergence Analysis
• N. Kumar *, R. Kumar Sinha, R. Ranjan Pages 711-727
The present study addresses an exponentially fitted finite difference method to obtain the solution of singularly perturbed two-point boundary value problems (BVPs) having a boundary layer at one end (left or right) point on uniform mesh. A fitting factor is introduced in the derived scheme using the theory of singular perturbations. Thomas algorithm is employed to solve the resulting tri-diagonal system of equations. The convergence of the presented method is investigated. Several model example problems are solved using the proposed method. The results are presented with terms of maximum absolute errors, which demonstrate the accuracy and efficiency of the method. It is observed that the proposed method is capable of producing highly accurate results with minimal computational effort for a fixed value of step size h, when the perturbation parameter tends to zero. From the graphs, we also observed that a numerical solution approximates the exact solution very well in the boundary layers for smaller value of ε.
Keywords: Singular perturbation problem, Stability, convergence, finite difference method
• S. Arora, I. Bala * Pages 728-746
The nonlinear space time dynamics have been discussed in terms of a hyper-bolic equation known as a sine-Gordon equation. The proposed equation has been discretized using the Bessel collocation method with Bessel poly-nomials as base functions. The proposed hyperbolic equation has been transformed into a system of parabolic equations using a continuously dif-ferentiable function. The system of equations involves one linear and the other nonlinear diffusion equation. The convergence of the present tech-nique has been discussed through absolute error, L2-norm, and L∞-norm.The numerical values obtained from the Bessel collocation method have been compared with the values already given in the literature. The present technique has been applied to different problems to check its applicability. Numerical values obtained from the Bessel collocation method have been presented in tabular as well as in graphical form.
Keywords: sine-Gordon equation, Bessel polynomials, Wave equation, Or-thogonal Collocation
• K. R. Cheneke * Pages 747-762
In this study, a modified model of HIV with therapeutic and preventive controls is developed. Moreover, a simple evaluation of the optimal control problem is investigated. We construct the Hamiltonian function by way of integrating Pontryagin’s maximal principle to achieve the point-wise optimal solution. The effects obtained from the version analysis strengthen public health education to a conscious population, PrEP for early activation of HIV infection prevention, and early treatment with artwork for safe life after HIV infection. Moreover, numerical simulations are done using the MATLAB platform to illustrate the qualitative conduct of the HIV infection. In the end, we receive that adhering to ART protective prone people, the usage of PrEP along with different prevention control is safer control measures.
Keywords: HIV, Optimal control problem, Basic reproduction number, Numerical simulation