This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We der...This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in detail, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory. .展开更多
This study presents a comprehensive numerical investigation of fourth-order nonlinear boundary value problems(BVPs)using an efficient and accurate computational approach.The present work focuses on a class of nonlinea...This study presents a comprehensive numerical investigation of fourth-order nonlinear boundary value problems(BVPs)using an efficient and accurate computational approach.The present work focuses on a class of nonlinear boundary value problems that commonly emerge in scientific and engineering applications,where the underlying models are often governed by complex nonlinear differential equations.Due to the difficulty of obtaining exact ana-lytical solutions for such problems,numerical techniques become essential for reliable approximation.In this work,the Finite Difference Method(FDM)is adopted as the core numerical tool due to its robustness and effectiveness in solving such problems.A carefully designed finite difference scheme is devel-oped to discretize the governing fourth-order nonlinear differential equations,converting them into a system of nonlinear algebraic equations.These systems are subsequently solved numerically using Maple software as the computa-tional tool.The article includes two illustrative examples of nonlinear BVPs to demonstrate the applicability and performance of the proposed method.Nu-merical results,including graphical representations,are provided for various step sizes.Both absolute and relative errors are calculated to assess the accu-racy of the solutions.The numerical findings are further validated by compar ing them with known analytical or previously published approximate results.The outcomes confirm that the finite difference approach yields highly accu-rate and reliable solutions.展开更多
In previous studies, the nonlinear problem of electrohydrodynamic(EHD)ion drag flows in a circular cylindrical conduit has been studied by several authors. However, those studies seldom involve the computation for lar...In previous studies, the nonlinear problem of electrohydrodynamic(EHD)ion drag flows in a circular cylindrical conduit has been studied by several authors. However, those studies seldom involve the computation for large physical parameters such as the electrical Hartmann number and the magnitude parameter for the strength of the nonlinearity due to the existence of strong nonlinearity in these extreme cases. To overcome this faultiness, the newly-developed homotopy Coiflets wavelet method is extended to solve this EHD flow problem with strong nonlinearity. The validity and reliability of the proposed technique are verified. Particularly, the highly accurate homotopy-wavelet solution is obtained for extreme large parameters, which seems to be overlooked before.Discussion about the effects of related physical parameters on the axial velocity field is presented.展开更多
The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linea...The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linear combination to the approximate solutions of 2nd order boundary value problems. Here the Bernoulli polynomials over the interval [0,1] are chosen as trial functions so that care has been taken to satisfy the corresponding homogeneous form of the Dirichlet boundary conditions in the Galerkin weighted residual method. In addition to that the given differential equation over arbitrary finite domain [a,b] and the boundary conditions are converted into its equivalent form over the interval [0,1]. All the formulas are verified by considering numerical examples. The approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. A reliable good accuracy is obtained in all cases.展开更多
In this paper, a new approach called Power Series Approximation Method (PSAM) is developed for the numerical solution of a generalized linear and non-linear higher order Boundary Value Problems (BVPs). The proposed me...In this paper, a new approach called Power Series Approximation Method (PSAM) is developed for the numerical solution of a generalized linear and non-linear higher order Boundary Value Problems (BVPs). The proposed method is efficient and effective on the experimentation on some selected thirteen-order, twelve-order and ten-order boundary value problems as compared with the analytic solutions and other existing methods such as the Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) available in the literature. A convergence analysis of PSAM is also provided.展开更多
In this article, the author employs the conical expansion and compression fixed point principle and the fixed point index theory to show that there exist at least two positive solutions for a higher order BVP.
A class of singularly perturbed nonlinear two-*point boundary value problems of ordinary differeotial equstions is studied. Under suitable monotonicity assumptions, we show the existence and uniqueness of solutions by...A class of singularly perturbed nonlinear two-*point boundary value problems of ordinary differeotial equstions is studied. Under suitable monotonicity assumptions, we show the existence and uniqueness of solutions by means of the homotopy approach and give an application in singular perturbations for optimal control problems.展开更多
文摘This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in detail, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory. .
文摘This study presents a comprehensive numerical investigation of fourth-order nonlinear boundary value problems(BVPs)using an efficient and accurate computational approach.The present work focuses on a class of nonlinear boundary value problems that commonly emerge in scientific and engineering applications,where the underlying models are often governed by complex nonlinear differential equations.Due to the difficulty of obtaining exact ana-lytical solutions for such problems,numerical techniques become essential for reliable approximation.In this work,the Finite Difference Method(FDM)is adopted as the core numerical tool due to its robustness and effectiveness in solving such problems.A carefully designed finite difference scheme is devel-oped to discretize the governing fourth-order nonlinear differential equations,converting them into a system of nonlinear algebraic equations.These systems are subsequently solved numerically using Maple software as the computa-tional tool.The article includes two illustrative examples of nonlinear BVPs to demonstrate the applicability and performance of the proposed method.Nu-merical results,including graphical representations,are provided for various step sizes.Both absolute and relative errors are calculated to assess the accu-racy of the solutions.The numerical findings are further validated by compar ing them with known analytical or previously published approximate results.The outcomes confirm that the finite difference approach yields highly accu-rate and reliable solutions.
基金the National Natural Science Foundation of China (No. 11872241)。
文摘In previous studies, the nonlinear problem of electrohydrodynamic(EHD)ion drag flows in a circular cylindrical conduit has been studied by several authors. However, those studies seldom involve the computation for large physical parameters such as the electrical Hartmann number and the magnitude parameter for the strength of the nonlinearity due to the existence of strong nonlinearity in these extreme cases. To overcome this faultiness, the newly-developed homotopy Coiflets wavelet method is extended to solve this EHD flow problem with strong nonlinearity. The validity and reliability of the proposed technique are verified. Particularly, the highly accurate homotopy-wavelet solution is obtained for extreme large parameters, which seems to be overlooked before.Discussion about the effects of related physical parameters on the axial velocity field is presented.
文摘The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linear combination to the approximate solutions of 2nd order boundary value problems. Here the Bernoulli polynomials over the interval [0,1] are chosen as trial functions so that care has been taken to satisfy the corresponding homogeneous form of the Dirichlet boundary conditions in the Galerkin weighted residual method. In addition to that the given differential equation over arbitrary finite domain [a,b] and the boundary conditions are converted into its equivalent form over the interval [0,1]. All the formulas are verified by considering numerical examples. The approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. A reliable good accuracy is obtained in all cases.
文摘In this paper, a new approach called Power Series Approximation Method (PSAM) is developed for the numerical solution of a generalized linear and non-linear higher order Boundary Value Problems (BVPs). The proposed method is efficient and effective on the experimentation on some selected thirteen-order, twelve-order and ten-order boundary value problems as compared with the analytic solutions and other existing methods such as the Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) available in the literature. A convergence analysis of PSAM is also provided.
基金the National Natural Science Foundation of China (No.19671052) and Postdoctorate Foundation of Zhengzhou Uniyersity.
文摘In this article, the author employs the conical expansion and compression fixed point principle and the fixed point index theory to show that there exist at least two positive solutions for a higher order BVP.
文摘A class of singularly perturbed nonlinear two-*point boundary value problems of ordinary differeotial equstions is studied. Under suitable monotonicity assumptions, we show the existence and uniqueness of solutions by means of the homotopy approach and give an application in singular perturbations for optimal control problems.
