Boundary Value Problems最新文献

{"title":"Enhanced shifted Jacobi operational matrices of derivatives: spectral algorithm for solving multiterm variable-order fractional differential equations","authors":"H. M. Ahmed","doi":"10.1186/s13661-023-01796-1","DOIUrl":"https://doi.org/10.1186/s13661-023-01796-1","url":null,"abstract":"Abstract This paper presents a new way to solve numerically multiterm variable-order fractional differential equations (MTVOFDEs) with initial conditions by using a class of modified shifted Jacobi polynomials (MSJPs). As their defining feature, MSJPs satisfy the given initial conditions. A key aspect of our methodology involves the construction of operational matrices (OMs) for ordinary derivatives (ODs) and variable-order fractional derivatives (VOFDs) of MSJPs and the application of the spectral collocation method (SCM). These constructions enable efficient and accurate numerical computation. We establish the error analysis and the convergence of the proposed algorithm, providing theoretical guarantees for its effectiveness. To demonstrate the applicability and accuracy of our method, we present five numerical examples. Through these examples, we compare the results obtained with other published results, confirming the superiority of our method in terms of accuracy and efficiency. The suggested algorithm yields very accurate agreement between the approximate and exact solutions, which are shown in tables and graphs.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134900899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Decay of the 3D Lüst model","authors":"Ying Sheng","doi":"10.1186/s13661-023-01797-0","DOIUrl":"https://doi.org/10.1186/s13661-023-01797-0","url":null,"abstract":"Abstract In this paper, we consider the time-decay rate of the strong solution to the Cauchy problem for the three-dimensional Lüst model. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained. The $dot{H}^{-s}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:math> ( $0leq s<frac{3}{2}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>s</mml:mi> <mml:mo><</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> ) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135138378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamical behavior of perturbed Gerdjikov–Ivanov equation through different techniques","authors":"Hamood Ur Rehman, Ifrah Iqbal, M. Mirzazadeh, Salma Haque, Nabil Mlaiki, Wasfi Shatanawi","doi":"10.1186/s13661-023-01792-5","DOIUrl":"https://doi.org/10.1186/s13661-023-01792-5","url":null,"abstract":"Abstract The objective of this work is to investigate the perturbed Gerdjikov–Ivanov (GI) equation along spatio-temporal dispersion which explains the dynamics of soliton dispersion and evolution of propagation distance in optical fibers, photonic crystal fibers (PCF), and metamaterials. The algorithms, namely hyperbolic extended function method and generalized Kudryashov’s method, are constructed to obtain the new soliton solutions. The dark, bright, periodic, and singular solitons are derived of the considered equation with the appropriate choice of parameters. These results are novel, confirm the stability of optical solitons, and have not been studied earlier. The explanation of evaluated results is given by sketching the various graphs in 3D, contour and 2D plots by using Maple 18. Graphical simulations divulge that varying the wave velocity affects the dynamical behaviors of the model. In summary, this research adds to our knowledge on how the perturbed GI equation with spatio-temporal dispersion behaves. The obtained soliton solutions and the methods offer computational tools for further analysis in this field. This work represents an advancement in our understanding of soliton dynamics and their applications in photonic systems.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence and multiplicity of solutions for the Cauchy problem of a fractional Lorentz force equation","authors":"Xiaohui Shen, Tiefeng Ye, Tengfei Shen","doi":"10.1186/s13661-023-01793-4","DOIUrl":"https://doi.org/10.1186/s13661-023-01793-4","url":null,"abstract":"Abstract This paper aims to deal with the Cauchy problem of a fractional Lorentz force equation. By the methods of reducing and topological degree in cone, the existence and multiplicity of solutions to the problem were obtained, which extend and enrich some previous results.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135813277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical solution of Bratu’s boundary value problem based on Green’s function and a novel iterative scheme","authors":"Junaid Ahmad, Muhammad Arshad, Kifayat Ullah, Zhenhua Ma","doi":"10.1186/s13661-023-01791-6","DOIUrl":"https://doi.org/10.1186/s13661-023-01791-6","url":null,"abstract":"Abstract We compute the numerical solution of the Bratu’s boundary value problem (BVP) on a Banach space setting. To do this, we embed a Green’s function into a new two-step iteration scheme. After this, under some assumptions, we show that this new iterative scheme converges to a sought solution of the one-dimensional non-linear Bratu’s BVP. Furthermore, we show that the suggested new iterative scheme is essentially weak $w^{2}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>w</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -stable in this setting. We perform some numerical computations and compare our findings with some other iterative schemes of the literature. Numerical results show that our new approach is numerically highly accurate and stable with respect to different set of parameters.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135411516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres","authors":"Kamal Ould Bouh","doi":"10.1186/s13661-023-01789-0","DOIUrl":"https://doi.org/10.1186/s13661-023-01789-0","url":null,"abstract":"Abstract This paper is devoted to studying the nonlinear problem with slightly subcritical and supercritical exponents $(S_{pm varepsilon}): Delta ^{2}u-c_{n}Delta u+d_{n}u = Ku^{ frac{n+4}{n-4}pm varepsilon}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>±</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>K</mml:mi> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:mfrac> <mml:mo>±</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msup> </mml:math> , $u&gt;0$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>u</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> on $S^{n}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , where $ngeq 5$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:math> , ε is a small positive parameter and K is a smooth positive function on $S^{n}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . We construct some solutions of $(S_{-varepsilon})$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:math> that blow up at one critical point of K . However, we prove also a nonexistence result of single-peaked solutions for the supercritical equation $(S_{+varepsilon})$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>+</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:math> .","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135411642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Determination of rigid inclusions immersed in an isotropic elastic body from boundary measurement","authors":"Mohamed Abdelwahed, Nejmeddine Chorfi, Maatoug Hassine","doi":"10.1186/s13661-023-01788-1","DOIUrl":"https://doi.org/10.1186/s13661-023-01788-1","url":null,"abstract":"Abstract We study the determination of some rigid inclusions immersed in an isotropic elastic medium from overdetermined boundary data. We propose an accurate approach based on the topological sensitivity technique and the reciprocity gap concept. We derive a higher-order asymptotic formula, connecting the known boundary data and the unknown inclusion parameters. The obtained formula is interesting and useful tool for developing accurate and robust numerical algorithms in geometric inverse problems.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136013010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law","authors":"Sabri T. M. Thabet, Thabet Abdeljawad, Imed Kedim, M. Iadh Ayari","doi":"10.1186/s13661-023-01790-7","DOIUrl":"https://doi.org/10.1186/s13661-023-01790-7","url":null,"abstract":"Abstract In this paper, new generalized weighted fractional derivatives with respect to another function are derived in the sense of Caputo and Riemann–Liouville involving a new modified version of a generalized Mittag–Leffler function with three parameters, as well as their corresponding fractional integrals. In addition, several new and existing operators of nonsingular kernels are obtained as special cases of our operator. Many important properties related to our new operator are introduced, such as a series version involving Riemann–Liouville fractional integrals, weighted Laplace transforms with respect to another function, etc. Finally, an example is given to illustrate the effectiveness of the new results.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On nonlinear fractional Choquard equation with indefinite potential and general nonlinearity","authors":"Fangfang Liao, Fulai Chen, Shifeng Geng, Dong Liu","doi":"10.1186/s13661-023-01786-3","DOIUrl":"https://doi.org/10.1186/s13661-023-01786-3","url":null,"abstract":"Abstract In this paper, we consider a class of fractional Choquard equations with indefinite potential $$ (-Delta )^{alpha}u+V(x)u= biggl[ int _{{mathbb{R}}^{N}} frac{M(epsilon y)G(u)}{ vert x-y vert ^{mu}},mathrm{d}y biggr]M( epsilon x)g(u), quad xin {mathbb{R}}^{N}, $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mo>∫</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:msub> <mml:mfrac> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mi>G</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:mi>y</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mi>μ</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>y</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>g</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:math> where $alpha in (0,1)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:math> , $N> 2alpha $ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>N</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> <mml:mi>α</mml:mi> </mml:math> , $0<mu <2alpha $ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>μ</mml:mi> <mml:mo><</mml:mo> <mml:mn>2</mml:mn> <mml:mi>α</mml:mi> </mml:math> , ϵ is a positive parameter. Here $(-Delta )^{alpha}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> </mml:math> stands for the fractional Laplacian, V is a linear potential with periodicity condition, and M is a nonlinear reaction potential with a global condition. We establish the existence and concentration of ground state solutions under general nonlinearity by using variational methods.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135435849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing Dirichlet eigenvalues of the Schrödinger operator with a PT-symmetric optical potential","authors":"Cemile Nur","doi":"10.1186/s13661-023-01787-2","DOIUrl":"https://doi.org/10.1186/s13661-023-01787-2","url":null,"abstract":"Abstract We provide estimates for the eigenvalues of non-self-adjoint Sturm–Liouville operators with Dirichlet boundary conditions for a shift of the special potential $4cos ^{2}x+4iVsin 2x$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>4</mml:mn> <mml:msup> <mml:mo>cos</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:mi>i</mml:mi> <mml:mi>V</mml:mi> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> <mml:mi>x</mml:mi> </mml:math> that is a PT-symmetric optical potential, especially when $|c|=|sqrt{1-4V^{2}}|<2$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>|</mml:mo> <mml:mi>c</mml:mi> <mml:mo>|</mml:mo> <mml:mo>=</mml:mo> <mml:mo>|</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> <mml:msup> <mml:mi>V</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msqrt> <mml:mo>|</mml:mo> <mml:mo><</mml:mo> <mml:mn>2</mml:mn> </mml:math> or correspondingly $0leq V<sqrt {5}/2$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>V</mml:mi> <mml:mo><</mml:mo> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> . We obtain some useful equations for calculating Dirichlet eigenvalues also for $|c|geq 2$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>|</mml:mo> <mml:mi>c</mml:mi> <mml:mo>|</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:math> or equally $Vgeq sqrt{5}/2$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>V</mml:mi> <mml:mo>≥</mml:mo> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> . We discuss our results by comparing them with the periodic and antiperiodic eigenvalues of the Schrödinger operator. We even approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error analysis.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135591274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}