A coupling method of a homotopy technique and a perturbation technique for nonlinear problems

DOI10.1016/S0020-7462(98)00085-7zbMath1068.74618OpenAlexW2087226000MaRDI QIDQ1609260

Ji-Huan He

Publication date: 15 August 2002

Published in: International Journal of Non-Linear Mechanics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/s0020-7462(98)00085-7

Mathematics Subject Classification ID

Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics (74H10) Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics (74G10)

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method coupled with the Fourier transform for nonlinear and singular Lane-Emden equations ⋮ Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals ⋮ Homotopy perturbation method: a new nonlinear analytical technique ⋮ A new iteration method for solving algebraic equations ⋮ Analytical solution of Abel integral equation arising in astrophysics via Laplace transform ⋮ Numerical solution of the singularly perturbed problem with nonlocal boundary condition ⋮ A note on delta-perturbation expansion method ⋮ Porosity reconstruction based on Biot elastic model of porous media by homotopy perturbation method ⋮ A study of moisture content in unsaturated porous medium by using homotopy perturbation method (HPM) and variational iteration method (VIM) ⋮ A computational algorithm for solution of population models for single and interacting species ⋮ Approximate solutions for higher order linear and nonlinear boundary value problems ⋮ Investigating the solutions of two classical nonlinear oscillators by the AG method ⋮ On higher-order differentiation in nonlinear mechanics ⋮ Asymptotic expansions for the Bessel transformations with high oscillations by using the homotopy technique ⋮ A spectral element approach for the stability of delay systems ⋮ Numerical solutions of the nonlinear integro-differential equations: wavelet-Galerkin method and homotopy perturbation method ⋮ Application of He's homotopy perturbation method to nonlinear integro-differential equations ⋮ Analytical solutions for the fractional nonlinear cable equation using a modified homotopy perturbation and separation of variables methods ⋮ Couple of the variational iteration method and Legendre wavelets for nonlinear partial differential equations ⋮ A new approach for solving fractional partial differential equations ⋮ Fixed-term homotopy ⋮ A generalized fractional sub-equation method for fractional differential equations with variable coefficients ⋮ ANALYTIC SOLUTIONS TO THE OSCILLATORY BEHAVIOR AND PRIMARY RESONANCE OF ELECTROSTATICALLY ACTUATED MICROBRIDGES ⋮ A new algorithm for solving differential equations ⋮ Compact structures in a class of nonlinearly dispersive equations with time-fractional derivatives ⋮ Homotopy analysis method for limit cycle flutter of airfoils ⋮ Numerical solution of two-dimensional nonlinear differential equation by homotopy perturbation method ⋮ A new modification of the homotopy perturbation method for linear and nonlinear operators ⋮ A numerical solution for solving system of Fredholm integral equations by using homotopy perturbation method ⋮ Application of He's homotopy perturbation method for \(n\)th-order integro-differential equations ⋮ A new family of iterative methods for solving system of nonlinear algebraic equations ⋮ Solutions of the Cahn-Hilliard equation ⋮ Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method ⋮ APPROXIMATING THE EFFECT OF VAN DER WAALS FORCE ON THE INSTABILITY OF ELECTROSTATIC NANO-CANTILEVERS ⋮ NUMERICAL SOLUTIONS OF WAVE EQUATIONS SUBJECT TO AN INTEGRAL CONSERVATION CONDITION BY HE'S HOMOTOPY PERTURBATION METHOD ⋮ Solutions of integral and integro-differential equation systems by using differential transform method ⋮ NUMERICAL SOLUTIONS FOR THE NONLINEAR FORNBERG–WHITHAM EQUATION BY HE'S METHODS ⋮ Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients ⋮ Approximate analytical solution to fractional modified KdV equations ⋮ Application of variational iteration method and homotopy perturbation method to the modified Kawahara equation ⋮ Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods ⋮ Series solutions for unsteady laminar MHD flow near forward stagnation point of an impulsively rotating and translating sphere in presence of buoyancy forces ⋮ Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel ⋮ A novel modeling methodology for memristive systems using homotopy perturbation methods ⋮ Homotopy analysis Sumudu transform method for time-fractional third order dispersive partial differential equation ⋮ A mathematical model on fractional Lotka-Volterra equations ⋮ Solving fractional-order diffusion equations in a plasma and fluids via a novel transform ⋮ Analytical and numerical calculation of heat transfer inside the hard nip calender ⋮ A novel approach for solving linear and nonlinear time-fractional Schrödinger equations ⋮ Uncertain dynamic responses of fuzzy fractionally damped spring-mass system ⋮ Analysis of fractional differential equations with the help of different operators ⋮ Analysis of fractional thin film flow of third grade fluid in lifting and drainage via homotopy perturbation procedure ⋮ Application of He's homotopy perturbation method to functional integral equations ⋮ Solutions of nonlinear oscillators by the modified differential transform method ⋮ Eventual periodicity of forced oscillations of the Korteweg-de Vries type equation ⋮ A spectral‐homotopy analysis method for heat transfer flow of a third grade fluid between parallel plates ⋮ Peaceman‐Rachford ADI scheme for the two dimensional flow of a second‐grade fluid ⋮ Application of He's homotopy perturbation method for multi‐dimensional fractional Helmholtz equation ⋮ Rational approximation solution of the foam drainage equation with time‐ and space‐fractional derivatives ⋮ Approximate solutions for Fornberg‐Whitham type equations ⋮ Homotopy analysis method for the one‐dimensional hyperbolic telegraph equation with initial conditions ⋮ An efficient method for Cauchy problem of ill‐posed nonlinear diffusion equation ⋮ Solution of the heat equation in the cast‐mould heterogeneous domain using a weighted algorithm based on the homotopy perturbation method ⋮ A novel Laplace decomposition method for non-linear stretching sheet problem in the presence of MHD and slip condition ⋮ Exact solutions of fractional heat-like and wave-like equations with variable coefficients ⋮ Fractional calculus for nanoscale flow and heat transfer ⋮ Application of the homotopy perturbation method to an inverse heat problem ⋮ Variational iteration method with He's polynomials for MHD Falkner-Skan flow over permeable wall based on Lie symmetry method ⋮ Approximate analytical solution for the one-dimensional nonlinear Boussinesq equation ⋮ On the solution of nonlinear vibration of truncated conical shells covered by functionally graded coatings ⋮ EFFECTS OF CASIMIR AND VAN DER WAALS FORCES ON THE PULL-IN INSTABILITY OF THE NONLINEAR MICRO AND NANO-BRIDGE GYROSCOPES ⋮ Analysis of the new homotopy perturbation method for linear and nonlinear problems ⋮ A comparison of two formulations of the fin efficiency for straight fins ⋮ Numerical solution of Rayleigh equation in non-linear vibration ⋮ On some modified variational iteration methods for solving the one-dimensional sine–Gordon equation ⋮ An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators ⋮ Homotopy analysis method for quadratic Riccati differential equation ⋮ MODIFIED HOMOTOPY PERTURBATION METHOD FOR SOLVING INTEGRAL EQUATIONS ⋮ Numerical solution for special nonlinear Fredholm integral equation by HPM ⋮ Homotopy perturbation method for thin film flow of a third-grade fluid down an inclined plane ⋮ On multistage homotopy-perturbation method applied to nonlinear biochemical reaction model ⋮ Asymptotic numerical method for multi-degree-of-freedom nonlinear dynamic systems ⋮ The secant-homotopy continuation method ⋮ Numerical solutions of highly oscillatory integrals ⋮ Nonlinear dynamic analysis of an unbalanced rotor supported by roller bearing ⋮ Bifurcation and stability of forced convection in tightly coiled ducts: multiplicity ⋮ Limit cycle and bifurcation of nonlinear problems ⋮ Application of homotopy perturbation method to nonlinear wave equations ⋮ Numerical solutions of the integral equations: homotopy perturbation method and Adomian's decomposition method ⋮ The inverse kinematics problem of spatial 4P3R robot manipulator by the homotopy continuation method with an adjustable auxiliary homotopy function ⋮ An analytical approximation for solving nonlinear Blasius equation by NHPM ⋮ The homotopy analysis method to solve the modified equal width wave equation ⋮ Application of adapted homotopy perturbation method for approximate solution of Henon-Heiles system ⋮ Comparison between variational iteration method and homotopy perturbation method for linear and nonlinear partial differential equations with the nonhomogeneous initial conditions ⋮ Analytic approximate solution of three-dimensional Navier-Stokes equations of flow between two stretchable disks ⋮ Application of homotopy perturbation method to multidimensional partial differential equations ⋮ A general approach to hyperbolic partial differential equations by homotopy perturbation method ⋮ Periodic solution of the Duffing-Van der Pol oscillator by homotopy perturbation method ⋮ Homotopy perturbation method to obtain exact special solutions with solitary patterns for Boussinesq-like B(m,n) equations with fully nonlinear dispersion ⋮ He's homotopy perturbation method for solving the space- and time-fractional telegraph equations ⋮ Application of multistage homotopy-perturbation method in hybrid synchronization of chaotic systems ⋮ Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations ⋮ Lie symmetry analysis to the time fractional generalized fifth-order KdV equation ⋮ A novel Lie-group theory and complexity of nonlinear dynamical systems ⋮ Generalized homotopy method for solving nonlinear differential equations ⋮ Invariant analysis and exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation by Lie group analysis ⋮ Homotopy perturbation method to obtain new solitary solutions with compact support for Boussinesq-like B(2n, 2n) equations with fully nonlinear dispersion ⋮ Homotopy perturbation method for a type of nonlinear coupled equations with parameters derivative ⋮ Rational approximation solution of the fractional Sharma-Tasso-Olever equation ⋮ Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order ⋮ Homotopy perturbation method for heat transfer flow of a third grade fluid between parallel plates ⋮ A study of convergence on the Newton-homotopy continuation method ⋮ Solvability of some fractional differential equations in the Hölder space \(\mathcal{H}_{\gamma}(\mathbb{R_+})\) and their numerical treatment via measures of noncompactness ⋮ Effect of wall conduction on steady natural convection Couette flow in a vertical channel ⋮ Effects of viscous and darcy dissipation on fully developed natural convection flow in a composite channel partially filled with porous material: Homotopy perturbation method (HPM) ⋮ Steady natural convection Couette flow with wall conduction and thermal boundary condition of third kind ⋮ Pattern formation in the models with coupling between order parameter and its gradient ⋮ An innovative efficient approach to solving damped Mathieu-Duffing equation with the non-perturbative technique ⋮ A novel Elzaki transform homotopy perturbation method for solving time-fractional non-linear partial differential equations ⋮ Homotopy perturbation and Adomian decomposition methods for condensing coagulation and Lifshitz-Slyzov models ⋮ An innovative technique to solve a fractal damping Duffing-jerk oscillator ⋮ Kashuri Fundo Decomposition Method for Solving Michaelis-Menten Nonlinear Biochemical Reaction Model ⋮ Unnamed Item ⋮ A variational approach to nonlinear two-point boundary value problems ⋮ A variational approach to nonlinear two-point boundary value problems ⋮ Application of He's homotopy perturbation and He's variational iteration methods for solution of Benney–Lin equation ⋮ Solution of the differential algebraic equations via homotopy perturbation method and their engineering applications ⋮ Solving singular nonlinear boundary value problems by combining the homotopy perturbation method and reproducing kernel Hilbert space method ⋮ On the approximation of integrals using homotopy perturbation method ⋮ LIE SYMMETRY ANALYSIS TO FISHER'S EQUATION WITH TIME FRACTIONAL ORDER ⋮ Analytical techniques for solving the equation governing the unsteady flow of a polytropic gas with time-fractional derivative ⋮ Iteration Perturbation Method for Strongly Nonlinear Oscillations ⋮ Application of the homotopy perturbation method for the Fokker-Planck equation ⋮ Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives ⋮ Approximations of the nonlinear Volterra's population model by an efficient numerical method ⋮ Analytical approach for solving population balances: a homotopy perturbation method ⋮ He's homotopy perturbation method for solving the shock wave equation ⋮ Transient Combustion Analysis of Iron Micro-Particles in a Gaseous Oxidizing Medium Using a New Iterative Method ⋮ Unnamed Item ⋮ Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer ⋮ He's MAX–MIN METHOD FOR THE RELATIVISTIC OSCILLATOR AND HIGH ORDER DUFFING EQUATION ⋮ Unnamed Item ⋮ Identification of diffusion parameters in a non-linear convection–diffusion equation using adaptive homotopy perturbation method ⋮ Comparing numerical methods for Boussinesq equation model problem ⋮ Exact solutions for nonlinear Burgers' equation by homotopy perturbation method ⋮ THE FRACTIONAL COMPLEX TRANSFORM: A NOVEL APPROACH TO THE TIME-FRACTIONAL SCHRÖDINGER EQUATION ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Application of homotopy perturbation method to solve two models of delay differential equation systems ⋮ The Homotopy Wiener-Hermite Expansion and Perturbation Technique (WHEP) ⋮ Unnamed Item ⋮ HE'S VARIATIONAL ITERATION METHOD FOR THE SOLUTION OF NONLINEAR NEWELL-WHITEHEAD-SEGEL EQUATION ⋮ 2-D Shallow Water Wave Equations with Fuzzy Parameters ⋮ On a symbolic method for neutral functional-differential equations with proportional delays ⋮ Homotopy perturbation method to fractional biological population equation ⋮ On partial fractional differential equations with variable coefficients ⋮ AN ELEMENTARY INTRODUCTION TO RECENTLY DEVELOPED ASYMPTOTIC METHODS AND NANOMECHANICS IN TEXTILE ENGINEERING ⋮ An efficient algorithm for solving nonlinear age-structured population models by combining homotopy perturbation and Padé techniques ⋮ Homotopy analysis method for solving fractional hyperbolic partial differential equations ⋮ Solution of Wave Equations and Heat Equations Using HPM ⋮ Series expansion of functions with He's homotopy perturbation method ⋮ A modified Newton–Raphson method ⋮ A meshless technique based on the radial basis functions for solving systems of partial differential equations ⋮ Higher-order homotopy perturbation method for conservative nonlinear oscillators generally and microelectromechanical systems’ oscillators particularly ⋮ Application of a class of iterative algorithms and their accelerations to Jacobian-based linearized EIT image reconstruction ⋮ A seminumeric approach for solution of the Eikonal partial differential equation and its applications ⋮ Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method ⋮ A new modified homotopy perturbation method with two free auxiliary parameters for solving MHD viscous flow due to a shrinking sheet ⋮ Direct solutions ofnth-order IVPs by homotopy-perturbation method ⋮ New modification of the HPM for numerical solutions of the sine-Gordon and coupled sine-Gordon equations ⋮ He's homotopy perturbation method for nonlinear differential-difference equations ⋮ Series solutions of a fractional oscillator by means of the homotopy perturbation method ⋮ A new analytical method for solving systems of Volterra integral equations ⋮ .An optimal Galerkin-homotopy asymptotic method applied to the nonlinear second-order BVPs ⋮ Numerical study of homotopy-perturbation method applied to Burgers equation in fluid ⋮ Non-perturbative solution of three-dimensional Navier-Stokes equations for the flow near an infinite rotating disk ⋮ The general analytical and numerical solution for the modified KdV equation with convergence analysis ⋮ Solving variational problems by homotopy–perturbation method ⋮ The homotopy analysis method for solving higher dimensional initial boundary value problems of variable coefficients ⋮ Homotopy perturbation method for numerical solutions of KdV‐Burgers' and Lax's seventh‐order KdV equations ⋮ A new analytical method for system of ODEs ⋮ He's homotopy perturbation method for fourth-order parabolic equations ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Application of the homotopy perturbation method to the modified regularized long-wave equation ⋮ A new HPM for ordinary differential equations ⋮ Application of HPM to the nonlinear vibrations of multiwalled carbon nanotubes ⋮ Variational iteration method for modified Camassa-Holm and Degasperis-Procesi equations ⋮ Bookkeeping Parameter in Perturbation Methods ⋮ Unnamed Item ⋮ ANALYTICAL COMPARISON BETWEEN THE HOMOTOPY PERTURBATION METHOD AND VARIATIONAL ITERATION METHOD FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER ⋮ Explicit analytical solutions of the generalized Burger and Burger-Fisher equations by homotopy perturbation method ⋮ Application of Homotopy Perturbation Method for Ecosystems Modelling ⋮ A reliable analytic study for higher-dimensional telegraph equation ⋮ Newton‐like iteration methods for solving non‐linear equations ⋮ Variational iteration method for solving three-dimensional Navier-Stokes equations of flow between two stretchable disks ⋮ Exact solutions for some systems of PDEs by He's homotopy perturbation method ⋮ Symbolic computation of exact solutions for fractional differential-difference equation models ⋮ Numerical simulations of the Boussinesq equation by He's variational iteration method ⋮ Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method ⋮ Unnamed Item ⋮ SOLUTIONS OF HELMHOLTZ EQUATIONS BY VARIATIONAL ITERATION METHOD ⋮ Series solution for a delay differential equation arising in electrodynamics ⋮ ANALYTICAL SOLUTION OF THE MAGNETO-HYDRODYNAMIC FLOW OVER A NONLINEAR STRETCHING SHEET ⋮ Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach ⋮ Unnamed Item ⋮ Periodic solution for strongly nonlinear vibration systems by He's variational iteration method ⋮ HE'S HOMOTOPY PERTURBATION METHOD FOR A GENERAL RICCATI EQUATION ⋮ Modified homotopy perturbation method for solving hypersingular integral equations of the second kind ⋮ On the solution of the nonlinear Kortewegâ de Vries equation by the homotopy perturbation method ⋮ Unnamed Item ⋮ Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method ⋮ Applications of variational iteration and homotopy perturbation methods to obtain exact solutions for time‐fractional diffusion‐wave equations ⋮ Homotopy perturbation method for numerical solutions of coupled Burgers equations with time‐ and space‐fractional derivatives ⋮ Analytical solution of wave system in Rⁿwith coupling controllers ⋮ Numerical soliton solution of the Kaup‐Kupershmidt equation ⋮ A numerical simulation of modified Boussinesq and seven‐order Sawada‐Kotara equations ⋮ The use of variational iteration method and homotopy perturbation method for solving two nonlinear equations ⋮ He's homotopy perturbation method for solving the fractional KdV‐Burgers‐Kuramoto equation ⋮ Unnamed Item ⋮ New Analytical Solution of Two-Dimensional Viscous Flow in a Rectangular Domain Bounded by Two Moving Porous Walls ⋮ Modified Variational Iteration Method for Traveling Wave Solutions of Seventh-order Generalized KdV Equations ⋮ Unnamed Item ⋮ Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Determination of natural frequencies by coupled method of homotopy perturbation and variational method for strongly nonlinear oscillators ⋮ EXACT AND APPROXIMATE ANALYTIC SOLUTIONS OF THE JEFFERY-HAMEL FLOW PROBLEM BY THE DUAN-RACH MODIFIED ADOMIAN DECOMPOSITION METHOD ⋮ An analytic approach to a class of fractional differential‐difference equations of rational type via symbolic computation ⋮ Relativistic effects on nonlinear lower hybrid oscillations in cold plasma ⋮ STUDY OF NONLINEAR HIROTA–SATSUMA COUPLED KdV AND COUPLED mKdV SYSTEM WITH TIME FRACTIONAL DERIVATIVE