University of TabrizComputational Methods for Differential Equations2345-398210420221001Numerical solution of the hyperbolic telegraph equation using cubic B-spline-based differential quadrature of high accuracy8378591415510.22034/cmde.2022.47744.1997ENAthiraBabuDepartment of Mathematics, Cochin University of Science and Technology, Kerala, India.BinHanDepartment of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada.NoufalAsharaf1Department of Mathematics, Cochin University of Science and Technology, Kerala, India.Journal Article20210902By constructing a newly modified cubic B-splines having the optimal accuracy order four, we propose a numerical scheme for solving the hyperbolic telegraph equation using a differential quadrature method. The spatial derivatives are approximated by the differential quadrature whose weight coefficients are computed using the newly modified cubic B-splines. Our modified cubic B-splines retain the tridiagonal structure and achieve the fourth order convergence rate. The solution of the associated ODEs is advanced in the time domain by the SSPRK scheme. The stability of the method is analyzed using the discretization matrix. Our numerical experiments demonstrate the better performance of our proposed scheme over several known numerical schemes reported in the literature.https://cmde.tabrizu.ac.ir/article_14155_ac0c9aa1a25cdf45a8878888044c9eee.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Sliding mode control of a class of uncertain nonlinear fractional-order time-varying delayed systems based on Razumikhin approach8608751392110.22034/cmde.2021.46019.1928ENMohammadGhamgosarDepartment of Mathematics, Payame Noor University (PNU), Tehran, Iran.Seyed MehdiMirhosseini-AlizaminiDepartment of Mathematics, Payame Noor University (PNU), Tehran, Iran.MahmoodDadkhahDepartment of Mathematics, Payame Noor University (PNU), Tehran, Iran.Journal Article20210511Within the current paper, we design a sliding-based control law to stabilize a set of systems that are nonlinear, fractional order involve delay, perturbation, and uncertainty. A control law-based sliding mode is considered in such a way that the variables of the closed loop system reach the sliding surface in a limited time and stay on it for later times. Then, using the Razomokhin stability theorem, the stability of the systems is proved and in the end, a calculation is found to search for useful methods. https://cmde.tabrizu.ac.ir/article_13921_a043ff9b86bddd922e4780e8557b7871.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001An epidemic model for drug addiction8768931414610.22034/cmde.2021.47532.1989ENRezaMemarbashiDepartment of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran.ZahraEbadiDepartment of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran.0000-0001-5578-9959Journal Article20210819The two most common ways to prevent spreading drug addiction are counseling and imprisonment. In this paper, we propose and study a model for the spread of drug addiction incorporating the effect of consultation and incarceration of addicted individuals. We extract the basic reproductive ratio and study the occurrence of backward bifurcation. Also, we study the local and global stability of drug-free and endemic equilibria under suitable conditions. Finally, we use numerical simulations to illustrate the obtained analytical results. https://cmde.tabrizu.ac.ir/article_14146_68d68f1e6ce17f41ee49a30635080471.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001A Bernoulli Tau method for numerical solution of feedback Nash differential games with an error estimation8949041418210.22034/cmde.2022.44213.1870ENMojtabaDehghan BanadakiDepartment of Applied Mathematics, Shahed University, Tehran, Iran.HamidrezaNavidiDepartment of Applied Mathematics, Shahed University, Tehran, Iran.0000-0003-1072-8786Journal Article20210127In the present study, an efficient combination of the Tau method with the Bernoulli polynomials is proposed for computing the Feedback Nash equilibrium in differential games over a finite horizon. By this approach, the system of Hamilton-Jacobi Bellman equations of a differential game derived from Bellman’s optimality principle is transferred to a nonlinear system of algebraic equations solvable by using Newton’s iteration method. Some illustrative examples are provided to show the accuracy and efficiency of the proposed numerical method.https://cmde.tabrizu.ac.ir/article_14182_a940b16312cc654ef875a7f08b66c243.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Optical Solitons and Rogue wave solutions of NLSE with variables coefficients and modulation instability analysis9059131423110.22034/cmde.2021.36045.1624ENEbruCavlak AslanDepartment of Mathematics, Science Faculty, Firat Universit, Elazığ, Turkey0000-0002-2291-4044MustafaIncDepartment of Computer Engineering, Biruni University, Istanbul, Turkey.Department of Medical Research, China Medical University Hospital, China Medical University,Taichung, Taiwan.Journal Article20191009In this work, we investigate soliton solutions of the generalized variable coefficients nonlinear Schr¨odinger equation. The Jacobi elliptic ansatz method is applied to obtain the optical soliton solutions. The necessary conditions that warrant the presence of these solutions are determined. We consider the Lie symmetry analysis of governing equation. Also, the stability of this equation is analyzed by the modulation instability. https://cmde.tabrizu.ac.ir/article_14231_7c3e0303a9cee14fe068434891e9c683.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Error analysis and Kronecker implementation of Chebyshev spectral collocation method for solving linear PDEs9149271393710.22034/cmde.2021.46776.1966ENMehdiRazaviDepartment of Applied Mathematics and Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran.Mohammad MehdiHosseiniDepartment of Applied Mathematics and Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran.AbbasSalemiDepartment of Applied Mathematics and Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran.0000-0002-9403-2550Journal Article20210702Numerical methods have essential role to approximate the solutions of Partial Differential Equations (PDEs). Spectral method is one of the best numerical methods of exponential order with high convergence rate to solve PDEs. In recent decades the Chebyshev Spectral Collocation (CSC) method has been used to approximate solutions of linear PDEs. In this paper, by using linear algebra operators, we implement Kronecker Chebyshev Spectral Collocation (KCSC) method for n-order linear PDEs. By statistical tools, we obtain that the Run times of KCSC method has polynomial growth, but the Run times of CSC method has exponential growth. Moreover, error upper bounds of KCSC and CSC methods are compared.https://cmde.tabrizu.ac.ir/article_13937_c9a28f5f8f6482c160b7bd7e23954249.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Study of the non-Newtonian behaviour of Reiner Rivlin relative to power law in arterial stenosis9289411453410.22034/cmde.2022.47241.1978ENNibeditaDashDepartment of Mathematics, School of Physical Sciences, Doon University, Dehradun, IndiaSaritaSinghDepartment of Mathematics, School of Physical Sciences, Doon University, Dehradun, IndiaJournal Article20210730The present paper develops the solution of steady axi-symmetric Navier-Stokes conservation equations incorporating Reiner Rivlin stress and strain rate relation that represents generalized non-Newtonian fluid. Perturbation solution is obtained to determine the flow field for axially symmetric stenosed artery. The flow field obtained from the Perturbation solution is compared with the exact analytical solution. In perturbation solution, cross viscosity that represents non Newtonian characteristics is considered a perturbation parameter, and the result obtained is observed to be dependent on the perturbation parameter. At smaller values of cross viscosity, the perturbation result is significantly closer to the analytical solution. But, as the values of cross viscosity increase, the perturbation results show a wider deviation from analytical results. Further, in this paper, the results of Reiner Rivlin are compared with the results obtained from the Power Law stress and strain rate relation. Such comparison of results of Reiner Rivlin with Power law is utilized to study the flow characteristics of blood. The flow profile in the case of Reiner Rivlin is observed to be significantly closer to that of Power law. The study infers that Reiner Rivlin’s constitutive relation is fairly suitable in simulating blood flow in arterial stenosis.https://cmde.tabrizu.ac.ir/article_14534_e83af580d381fdf6663131f6e60a43dc.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Combining the reproducing kernel method with a practical technique to solve the system of nonlinear singularly perturbed boundary value problems9429531297610.22034/cmde.2021.40288.1758ENSaeidAbbasbandyDepartment of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin 34149-16818, Iran.HusseinSahihiDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.TofighAllahviranlooBahcesehir University, Faculty of Engineering and Natural Sciences, Istanbul, Turkey.0000-0002-6673-3560Journal Article20200611In this paper, a reliable new scheme is presented based on combining Reproducing Kernel Method (RKM) with a practical technique for the nonlinear problem to solve the System of Singularly Perturbed Boundary Value Problems (SSPBVP). The Gram-Schmidt orthogonalization process is removed in the present RKM. However, we provide error estimation for the approximate solution and its derivative. Based on the present algorithm in this paper, can also solve linear problem. Several numerical examples demonstrate that the present algorithm does have higher precision. https://cmde.tabrizu.ac.ir/article_12976_2413349b474d733e951ce18a8f32e8e3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Parameter-uniform robust scheme for singularly perturbed parabolic convection-diffusion problems with large time-lag9549681446210.22034/cmde.2022.47907.2006ENNaol TufaNegeroDepartment of Mathematics, Wollega University, Nekemte, Ethiopia.0000-0003-1593-735XGemechis FileDuressaDepartment of Mathematics, Jimma University, Jimma, Ethiopia.0000-0003-1889-4690Journal Article20210913This paper deals with a parameter uniform numerical method for singularly perturbed time delayed parabolic convection-diffusion problems. The method consists of a backward-Euler to discretize in temporal dimension and exponentially fitted B-spline collocation scheme for the spatial dimension on a uniform mesh. Parameter-uniform error estimates are obtained, and the method is proved uniformly convergent. The developed scheme is tested on various problems and observed to support the theoretical results. Finally, the numerical solutions are compared with the existing literature methods, and the present method is more accurate.https://cmde.tabrizu.ac.ir/article_14462_0ecf4acd0f1a34e47bf9151150cc5479.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001A numerical solution of two-dimensional hyperbolic telegraph equation based on moving least square meshless method and radial basis functions9699851334110.22034/cmde.2021.42440.1829ENSepidehNiknamDepartment of Applied Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.0000-0003-2579-1913HojatollahAdibiDepartment of Applied Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.0000-0002-8167-5020Journal Article20201023In this research, a linear combination of moving least square (MLS) and local radial basis functions (LRBFs) is considered within the framework of the meshless method to solve the two-dimensional hyperbolic telegraph equation. Besides, the differential quadrature method (DQM) is employed to discretize temporal derivatives. Furthermore, a control parameter is introduced and optimized to achieve minimum errors via an experimental approach. Illustrative examples are provided to demonstrate the applicability and efficiency of the method. The results prove the superiority of this method over using MLS and LRBF individually. https://cmde.tabrizu.ac.ir/article_13341_1646dd14e487b5950137cd8ab88da39f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Numerical analysis of fluid flow behaviour in two sided deep lid driven cavity using the finite volume technique98610061443810.22034/cmde.2022.49864.2074ENManoj RPatelDepartment of Mathematics, LDRP Institute of Technology and Research, Kadi Sarva Vishwavidyalaya, Gandhinagar, Gujarat, India.0000-0003-1534-7771Jigisha U.PandyaDepartment of Mathematics, Sarvajanik College of Engineering and Technology, Surat, Gujarat Technological University, Gujarat, India.Vijay KPatelDepartment of Mathematics, LDRP Institute of Technology and Research, Kadi Sarva Vishwavidyalaya, Gandhinagar, Gujarat, India.Journal Article20220111In the present study, numerical simulations of two-dimensional steady-state incompressible Newtonian fluid flow in one-sided square and two-sided deep lid driven cavities under the aspect ratio K = 1, 4, 6 are reported. For the one-sided lid driven cavity, the upper wall is moved to the right with up to 5000 Reynolds numbers under a grid size of up to 501×501. This lends support to previous findings in the literature with Ghia et al.s results. Three cases are used in this article for the two-sided deep lid driven square cavity specifically. In these cases, the top and lower walls are moved to the right, while the left and right walls remain fixed up to at high Reynolds numbers (5000) under the grid size of up to 201×201. All possible flow solutions are studied in the present article, and flow bifurcation diagrams are constructed as velocity profiles and streamline contours for the same Reynolds number using a finite volume SIMPLE technique. The work done in this paper includes flow properties such as the location of primary and secondary vortices, velocity components, and numerical values for benchmarking purposes, and it is in excellent agreement with previous findings in the literature. A PARAM Shavak, high-performance computing (HPC) computer, was used to execute the calculations.https://cmde.tabrizu.ac.ir/article_14438_4be04f8d30e88c6b2ffba69ba0bf6cee.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001A two-step method adaptive with memory with eighth-order for solving nonlinear equations and its dynamic100710261415310.22034/cmde.2022.46651.1961ENValiTorkashvandDepartment of Mathematics, Shahre-Qods Branch, Islamic Azad University, Tehran, Iran.Department of Basic Science, Shahid Sattari Aeronatuical Uinversity of Science and Technology, Tehran, Iran.0000-0001-8033-8279Journal Article20210622In this work, we have constructed the with memory two-step method with four convergence degrees by entering the maximum self-accelerator parameter(three parameters). Then, using Newton’s interpolation, a with-memory method with a convergence order of 7.53 is constructed. Using the information of all the steps, we will improve the convergence order by one hundred percent, and we will introduce our method with convergence order 8. Numerical examples demonstrate the exceptional convergence speed of the proposed method and confirm theoretical results. Finally, we have presented the dynamics of the adaptive method and other without-memory methods for complex polynomials of degrees two, three, and four. The basins of attraction of existing with-memory methods are present and compared to illustrate their performance.https://cmde.tabrizu.ac.ir/article_14153_d220454953782bfebf8ee9950f03bae3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001An optimal B-spline collocation technique for numerical simulation of viscous coupled Burgers’ equation102710451418110.22034/cmde.2021.46178.1936ENShallu.Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, India.Vijay KumarKukrejaDepartment of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, India.0000-0001-9717-9526Journal Article20210521In this paper, an optimal cubic B-spline collocation method is applied to solve the viscous coupled Burgers’ equation, which helps in modeling the polydispersive sedimentation. As it is not possible to obtain optimal order of convergence with the standard collocation method, so to overcome this, posteriori corrections are made in cubic B-spline interpolant and its higher-order derivatives. This optimal cubic B-spline collocation method is used for space integration and for time-domain integration, the Crank-Nicolson scheme is applied along with the quasilinearization process to deal with the nonlinear terms in the equations. Von-Neumann stability analysis is carried out to discuss the stability of the technique. Few test problems are solved numerically along with the calculation of L2, L∞ error norms as well as the order of convergence. The obtained results are compared with those available in the literature, which shows the improvement in results over the standard collocation method and many other existing techniques also.https://cmde.tabrizu.ac.ir/article_14181_17ec4fae0d735d8a1f1a4931a08588af.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001A novel perspective for simulations of the MEW equation by trigonometric cubic B-spline collocation method based on Rubin-Graves type linearization104610581414810.22034/cmde.2021.47358.1981ENNuri MuratYağmurluInönü University, Department of Mathematics, Malatya, 44280, Turkey.0000-0003-1593-0254Ali SercanKarakaşInönü University, Department of Mathematics, Malatya, 44280, Turkey.0000-0001-8622-1127Journal Article20210808In the present study, the Modified Equal Width (MEW) wave equation is going to be solved numerically by presenting a new technique based on the collocation finite element method in which trigonometric cubic B-splines are used as approximate functions. In order to support the present study, three test problems; namely, the motion of a single solitary wave, the interaction of two solitary waves, and the birth of solitons are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the numerical conserved laws as well as the error norms L2 and L∞. https://cmde.tabrizu.ac.ir/article_14148_ecd247d6531f74dc7d1daca03a421df3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001On the numerical scheme for solving non-linear two-dimensional Hammerstein integral equations105910741420810.22034/cmde.2022.47106.1974ENManochehrKazemiDepartment of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, Iran.Mohammad RezaDoostdarDepartment of mathematics, Zarandieh Branch, Islamic Azad University, Zarandieh, Iran.Journal Article20210721In this work, solving non-linear two-dimensional Hammerstein integral equations is considered by an iterative method of successive approximation. This method is an efficient approach based on a combination of the quadrature formula and the successive approximations method. Also, the convergence analysis and the numerical stability of the suggested method are studied. Finally, to survey the accuracy of the present method, some numerical experiments are given. https://cmde.tabrizu.ac.ir/article_14208_ad6012e8987b76fb0a3d91a7221634af.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Approximate price of the option under discretization by applying fractional quadratic interpolation107510851387310.22034/cmde.2021.45656.1918ENHamidMesgaraniDepartment of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785
-136, Iran.AdeleAdlDepartment of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785
-136, Iran.YonesEsmaeelzade AghdamDepartment of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785
-136, Iran.Journal Article20210424The time-fractional Black-Scholes model (TFBSM) governing European options in which the temporal derivative is focused on the Caputo fractional derivative with 0 < β ≤ 1 is considered in this article. Approximating financial options with respect to their hereditary characteristics can be well understood and explained due to its outstanding memory effect current in fractional derivatives. Compelled by the stated cause, It is important to find reasonably accurate and successful numerical methods when approaching fractional differential equations. The simulation model given here is developed in two ways: one, the semi-discrete is produced in the time using a quadratic interpolation with the order of precision τ3−α in the case of a smooth solution, and subsequently, the unconditional stability and convergence order are investigated. The spatial derivative variables are simulated using the collocation approach based on a Legendre basis for the designed full-discrete scheme. Last, we employ various test problems to demonstrate the suggested design’s high precision. Moreover, the obtained results are compared to those obtained using other methodologies, demonstrating that the proposed technique is highly accurate and practicable.https://cmde.tabrizu.ac.ir/article_13873_82765fbc2793e24c6ee53d1d7b13770a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Numerical solution of Drinfel’d–Sokolov system with the Haar wavelets method108610961415710.22034/cmde.2022.47018.1972ENSahbaHeydaryFaculty of Mathematics, K. N. Toosi University of Technology, P. O. Box: 16765–3381, Tehran, Iran.AzimAminataeiFaculty of Mathematics, K. N. Toosi University of Technology, P. O. Box: 16765–3381, Tehran, Iran.0000-0001-5247-4492Journal Article20210717In this article, we use the Haar wavelets (HWs) method to numerically solve the nonlinear Drinfel’d–Sokolov (DS) system. For this purpose, we use an approximation of functions with the help of HWs, and we approximate spatial derivatives using this method. In this regard, to linearize the nonlinear terms of the equations, we use the quasilinearization technique. At the end, to show the effectiveness and accuracy of the method in solving this system one numerical example is provided. https://cmde.tabrizu.ac.ir/article_14157_7bc257a38e0586735e7bbd23f7d05ee2.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001A numerical scheme for solving time-fractional Bessel differential equations109711141387410.22034/cmde.2021.45407.1910ENSaberTavanDepartment of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.MohammadJahangiri RadDepartment of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.AliSalimi ShamlooDepartment of Mathematics, Shabestar Branch, Islamic Azad
University, Shabestar, Iran.YaghoubMahmoudiDepartment of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.Journal Article20210412The object of this paper devotes on offering an indirect scheme based on time-fractional Bernoulli functions in the sense of Rieman-Liouville fractional derivative which ends up to the high credit of the obtained approximate fractional Bessel solutions. In this paper, the operational matrices of fractional Rieman-Liouville integration for Bernoulli polynomials are introduced. Utilizing these operational matrices along with the properties of Bernoulli polynomials and the least squares method, the fractional Bessel differential equation converts into a nonlinear system of algebraic. To solve these nonlinear algebraic equations which are a prominent the problem, there is a need to employ Newton’s iterative method. In order to elaborate the study, the synergy of the proposed method is investigated and then the accuracy and the efficiency of the method are clearly evaluated by presenting numerical results.https://cmde.tabrizu.ac.ir/article_13874_e7a8e37f234701bfe64d54866906afe5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001An interval chaos insight to iterative decomposition method for Rossler differential equation by considering stable uncertain coefficients111511221426910.22034/cmde.2021.20079.1215ENMajidAbbasiDepartment of Electrical Engineering, Tafresh University, Tafresh 39518-79611, Iran.MehdiRamezaniDepartments of Mathematics, Tafresh University, Tafresh 39518-79611, Iran.0000-0003-2495-7758Journal Article20170414Generally, in most applications of engineering, the parameters of the mathematical models are considered deterministic. Although, in practice, there are always some uncertainties in the model parameters; these uncertainties may be made wrong representation of the mathematical model of the system. These uncertainties can be generated from different reasons like measurement error, inhomogeneity of the process, chaotic behavior of systems, etc. This problem leads researchers to study these uncertainties and propose solutions for this problem. The iterative analysis is a method that can be utilized to solve these kinds of problems. In this paper, a new combined method based on interval chaotic and iterative decomposition method is proposed. The validation of the proposed method is performed on a chaotic Rossler system in stable Intervals. The simulation results are applied on 2 practical case studies and the results are compared with the interval Chebyshev method and RungeKutta method of order four (RK4) method. The final results showed that the proposed method has a good performance in finding the confidence interval for the Rossler models with interval uncertainties; the results also showed that the proposed method can handle the wrapping effect in a better manner to sharpen the range of non-monotonic interval.https://cmde.tabrizu.ac.ir/article_14269_cc703ab0f1b4a999d06071ef8f7a5daf.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398210420221001Asymptotic method for solution of oscillatory fractional derivative112311301386010.22034/cmde.2021.46895.1973ENFikret AhmadaliAlievInstitute of Applied Mathematics, Baku State University, Baku, Azerbaijan.Institute of Information Technologies of the National Academy of Sciences of Azerbaijan.NihanAlievInstitute of Applied Mathematics, Baku State University, Baku, Azerbaijan.NazileHajiyevaInstitute of Applied Mathematics, Baku State University, Baku, Azerbaijan.NargizSafarovaInstitute of Applied Mathematics, Baku State University, Baku, Azerbaijan.RufanaAliyevaNakhchivan State University, Nakhchivan, Azerbaijan.Journal Article20210717In the paper, an oscillatory system with liquid dampers is considered, when the mass of the head is large enough. By means of expedient transformations, the equation of motion with fractional derivatives is reduced to an equation of fractional order containing a small parameter. The corresponding nonlocal boundary value problem is solved and the zero and first approximations of solutions of the relative small parameter are constructed. The results are illustrated on the concrete example, where the solution differs from the analytical solution by 10−2 order. https://cmde.tabrizu.ac.ir/article_13860_aa1be6fdc00c4f017daae4119a671464.pdf