Applied Mathematics and Computation, ISSN 0096-3003, 12/2017, Volume 315, pp. 591 - 602

In this article, a new method is generated to solve nonlinear Lane–Emden type equations using Legendre, Hermite and Laguerre wavelets. We are interested to...

Isomorphism | Legendre wavelet | Hermite wavelet | Lane–Emden type equations | Collocation method | Laguerre wavelet | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | Lane-Emden type equations | NUMERICAL-SOLUTION | ALGORITHM | DIFFERENTIAL-EQUATIONS | INITIAL-VALUE PROBLEMS | Analysis | Methods | Differential equations

Isomorphism | Legendre wavelet | Hermite wavelet | Lane–Emden type equations | Collocation method | Laguerre wavelet | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | Lane-Emden type equations | NUMERICAL-SOLUTION | ALGORITHM | DIFFERENTIAL-EQUATIONS | INITIAL-VALUE PROBLEMS | Analysis | Methods | Differential equations

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 02/2015, Volume 275, pp. 321 - 334

Klein/Sine-Gordon equations are very important in that they can accurately model many essential physical phenomena. In this paper, we propose a new spectral...

Legendre wavelets | Klein[formula omitted]Sine-Gordon equation | Spectral method | Hierarchical scale structure | Klein{set minus}Sine-Gordon equation | Klein/Sine-Gordon equation | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | Klein\Sine-Gordon equation

Legendre wavelets | Klein[formula omitted]Sine-Gordon equation | Spectral method | Hierarchical scale structure | Klein{set minus}Sine-Gordon equation | Klein/Sine-Gordon equation | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | Klein\Sine-Gordon equation

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 09/2016, Volume 40, Issue 17-18, pp. 8087 - 8107

•We define a new fractional function based on the Bernoulli wavelet.•The fractional integration operational matrix for these functions is driven.•This matrix...

Operational matrix | Fractional-order Bernoulli wavelet | Bernoulli wavelet | Fractional differential equations | Numerical solution | Caputo derivative | INTEGRODIFFERENTIAL EQUATIONS | FORMULA | HOMOTOPY ANALYSIS METHOD | POLYNOMIALS | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | INTEGRATION | PARTIAL-DIFFERENTIAL-EQUATIONS | SYSTEMS | LEGENDRE FUNCTIONS | DERIVATIVES | Differential equations

Operational matrix | Fractional-order Bernoulli wavelet | Bernoulli wavelet | Fractional differential equations | Numerical solution | Caputo derivative | INTEGRODIFFERENTIAL EQUATIONS | FORMULA | HOMOTOPY ANALYSIS METHOD | POLYNOMIALS | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | INTEGRATION | PARTIAL-DIFFERENTIAL-EQUATIONS | SYSTEMS | LEGENDRE FUNCTIONS | DERIVATIVES | Differential equations

Journal Article

Physics Letters A, ISSN 0375-9601, 01/2015, Volume 379, Issue 3, pp. 71 - 76

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) is proposed for solving the time fractional diffusion-wave...

Riemann–Liouville integral | Hat functions (HFs) | Fractional operational matrix (FOM) | Legendre wavelets (LWs) | Caputo derivative | Fractional diffusion-wave equation (FDWE) | Caputo derivative Riemann-Liouville integral | 2-DIMENSIONAL LEGENDRE WAVELETS | PHYSICS, MULTIDISCIPLINARY | NUMERICAL-METHOD | STABILITY | INTEGRAL-EQUATIONS | OPERATIONAL MATRIX | SCHEME | FINITE-DIFFERENCE METHODS | Riemann-Liouville integral

Riemann–Liouville integral | Hat functions (HFs) | Fractional operational matrix (FOM) | Legendre wavelets (LWs) | Caputo derivative | Fractional diffusion-wave equation (FDWE) | Caputo derivative Riemann-Liouville integral | 2-DIMENSIONAL LEGENDRE WAVELETS | PHYSICS, MULTIDISCIPLINARY | NUMERICAL-METHOD | STABILITY | INTEGRAL-EQUATIONS | OPERATIONAL MATRIX | SCHEME | FINITE-DIFFERENCE METHODS | Riemann-Liouville integral

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 08/2016, Volume 286, pp. 139 - 154

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) is proposed for solving a class of fractional optimal...

Fractional optimal control problems | Hat functions (HFs) | Operational matrices | Legendre wavelets (LWs) | Lagrange multipliers method | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | DIFFERENTIAL-EQUATIONS | SYSTEMS | INTEGRODIFFERENTIAL EQUATIONS | Algebra | Computation | Lagrange multipliers | Mathematical analysis | Nonlinearity | Mathematical models | Dynamical systems | State variable

Fractional optimal control problems | Hat functions (HFs) | Operational matrices | Legendre wavelets (LWs) | Lagrange multipliers method | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | DIFFERENTIAL-EQUATIONS | SYSTEMS | INTEGRODIFFERENTIAL EQUATIONS | Algebra | Computation | Lagrange multipliers | Mathematical analysis | Nonlinearity | Mathematical models | Dynamical systems | State variable

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 2011, Volume 62, Issue 3, pp. 1038 - 1045

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order...

Legendre wavelet | Fractional differential equations | Numerical solution | Caputo fractional derivative | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | NUMERICAL-SOLUTIONS | TRANSFORM METHOD | HOMOTOPY PERTURBATION METHOD | Wavelet | Approximation | Computer simulation | Mathematical analysis | Differential equations | Exact solutions | Mathematical models | Derivatives

Legendre wavelet | Fractional differential equations | Numerical solution | Caputo fractional derivative | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | NUMERICAL-SOLUTIONS | TRANSFORM METHOD | HOMOTOPY PERTURBATION METHOD | Wavelet | Approximation | Computer simulation | Mathematical analysis | Differential equations | Exact solutions | Mathematical models | Derivatives

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 05/2014, Volume 234, pp. 267 - 276

In this paper, a new method based on the Legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis...

Operational matrices | Fractional partial differential equations | Two-dimensional Legendre wavelets | FOKKER-PLANCK EQUATION | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | APPROXIMATIONS | SYSTEMS | Methods | Differential equations | Wavelet | Partial differential equations | Matrices (mathematics) | Mathematical analysis | Dirichlet problem | Boundary conditions | Derivatives | Convergence

Operational matrices | Fractional partial differential equations | Two-dimensional Legendre wavelets | FOKKER-PLANCK EQUATION | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | APPROXIMATIONS | SYSTEMS | Methods | Differential equations | Wavelet | Partial differential equations | Matrices (mathematics) | Mathematical analysis | Dirichlet problem | Boundary conditions | Derivatives | Convergence

Journal Article

Asian Journal of Control, ISSN 1561-8625, 09/2018, Volume 20, Issue 5, pp. 1804 - 1817

In this paper, a new computational method based on the Legendre wavelets (LWs) is proposed for solving a class of variable‐order fractional optimal control...

Lagrange multiplier method | hat functions (HFs) | operational matrix of variable‐order fractional integration (OMV‐FI) | Legendre wavelets (LWs) | Variable‐order fractional optimal control problems (V‐FOCPs) | operational matrix of variable-order fractional integration (OMV-FI) | Variable-order fractional optimal control problems (V-FOCPs) | 2-DIMENSIONAL LEGENDRE WAVELETS | DIFFERENTIAL-EQUATIONS | CONSTANT-ORDER | OPERATIONAL MATRIX | ANOMALOUS DIFFUSION | NUMERICAL-SOLUTION | COMPUTATIONAL METHOD | AUTOMATION & CONTROL SYSTEMS | Analysis | Methods | Differential equations

Lagrange multiplier method | hat functions (HFs) | operational matrix of variable‐order fractional integration (OMV‐FI) | Legendre wavelets (LWs) | Variable‐order fractional optimal control problems (V‐FOCPs) | operational matrix of variable-order fractional integration (OMV-FI) | Variable-order fractional optimal control problems (V-FOCPs) | 2-DIMENSIONAL LEGENDRE WAVELETS | DIFFERENTIAL-EQUATIONS | CONSTANT-ORDER | OPERATIONAL MATRIX | ANOMALOUS DIFFUSION | NUMERICAL-SOLUTION | COMPUTATIONAL METHOD | AUTOMATION & CONTROL SYSTEMS | Analysis | Methods | Differential equations

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 2006, Volume 181, Issue 2, pp. 1417 - 1422

A numerical method for solving the Lane–Emden equations as singular initial value problems is presented. Using integral operator and convert Lane–Emden...

Legendre wavelets | Lane–Emden equations | Integral equations | Gaussian integration | Lane-Emden equations | integral equations | MATHEMATICS, APPLIED

Legendre wavelets | Lane–Emden equations | Integral equations | Gaussian integration | Lane-Emden equations | integral equations | MATHEMATICS, APPLIED

Journal Article

Nonlinear Dynamics, ISSN 0924-090X, 7/2016, Volume 85, Issue 2, pp. 1185 - 1202

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) together with the Galerkin method is proposed for solving a...

Brownian motion process | Duffing–Van der Pol Oscillator | Stochastic Brusselator problem | Nonlinear stochastic Itô–Volterra integral equations | Engineering | Vibration, Dynamical Systems, Control | Legendre wavelets (LWs) | Stochastic volatility models | Mechanics | Automotive Engineering | Mechanical Engineering | Stochastic Lotka–Volterra model | Stochastic operational matrix (SOM) | INTEGRODIFFERENTIAL EQUATIONS | TIME | ENGINEERING, MECHANICAL | Stochastic Lotka-Volterra model | Duffing-Van der Pol Oscillator | OPERATIONAL MATRIX | Nonlinear stochastic Ito-Volterra integral equations | NUMERICAL-SOLUTION | MECHANICS | RANDOM DIFFERENTIAL-EQUATIONS | COMPUTATIONAL METHOD | VOLTERRA-EQUATIONS | Organic chemistry | Nonlinear equations | Basis functions | Integral equations | Collocation methods | Wavelet analysis | Galerkin method | Nonlinear systems | Matrix methods | Volterra integral equations

Brownian motion process | Duffing–Van der Pol Oscillator | Stochastic Brusselator problem | Nonlinear stochastic Itô–Volterra integral equations | Engineering | Vibration, Dynamical Systems, Control | Legendre wavelets (LWs) | Stochastic volatility models | Mechanics | Automotive Engineering | Mechanical Engineering | Stochastic Lotka–Volterra model | Stochastic operational matrix (SOM) | INTEGRODIFFERENTIAL EQUATIONS | TIME | ENGINEERING, MECHANICAL | Stochastic Lotka-Volterra model | Duffing-Van der Pol Oscillator | OPERATIONAL MATRIX | Nonlinear stochastic Ito-Volterra integral equations | NUMERICAL-SOLUTION | MECHANICS | RANDOM DIFFERENTIAL-EQUATIONS | COMPUTATIONAL METHOD | VOLTERRA-EQUATIONS | Organic chemistry | Nonlinear equations | Basis functions | Integral equations | Collocation methods | Wavelet analysis | Galerkin method | Nonlinear systems | Matrix methods | Volterra integral equations

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 2011, Volume 16, Issue 11, pp. 4163 - 4173

► We solve fractional differential equations by wavelets. ► An operational matrix is derived. ► Initial and boundary value problems are solved. Fractional...

Fractional differential equations | Legendre wavelets | Operational matrices | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | BOUNDARY-VALUE PROBLEM | PHYSICS, FLUIDS & PLASMAS | INTEGRAL-EQUATIONS | PHYSICS, MATHEMATICAL | Wavelet | Algebra | Computer simulation | Mathematical analysis | Differential equations | Nonlinearity | Mathematical models

Fractional differential equations | Legendre wavelets | Operational matrices | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | BOUNDARY-VALUE PROBLEM | PHYSICS, FLUIDS & PLASMAS | INTEGRAL-EQUATIONS | PHYSICS, MATHEMATICAL | Wavelet | Algebra | Computer simulation | Mathematical analysis | Differential equations | Nonlinearity | Mathematical models

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 08/2015, Volume 46, pp. 83 - 88

In this paper, a numerical method is proposed to solve a class of nonlinear variable order fractional differential equations (FDEs). The idea is to use...

Legendre wavelets | Variable order fractional differential equations | Caputo fractional derivatives | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | INTEGRODIFFERENTIAL EQUATIONS | Differential equations | Numerical Analysis | Mathematics

Legendre wavelets | Variable order fractional differential equations | Caputo fractional derivatives | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | INTEGRODIFFERENTIAL EQUATIONS | Differential equations | Numerical Analysis | Mathematics

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 04/2015, Volume 256, pp. 715 - 723

In this paper, Legendre wavelet method is developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. The properties...

Legendre wavelets | Integro-differential equations | Legendre wavelet method | System of nonlinear Volterra integral equations | MATRIX | MATHEMATICS, APPLIED | MODEL | Methods | Differential equations

Legendre wavelets | Integro-differential equations | Legendre wavelet method | System of nonlinear Volterra integral equations | MATRIX | MATHEMATICS, APPLIED | MODEL | Methods | Differential equations

Journal Article

Composite Structures, ISSN 0263-8223, 08/2015, Volume 126, pp. 227 - 232

The accuracy issues of Haar wavelet method are studied. The order of convergence as well as error bound of the Haar wavelet method is derived for general nth...

Numerical evaluation of the order of convergence | Haar wavelet method | Extrapolation | Convergence theorem | Accuracy issues | NUMERICAL-SOLUTION | 2-DIMENSIONAL LEGENDRE WAVELETS | PARTIAL-DIFFERENTIAL-EQUATIONS | INTEGRAL-EQUATIONS | MATERIALS SCIENCE, COMPOSITES | SHELLS | FREE-VIBRATION ANALYSIS | Wavelet | Accuracy | Theorems | Discretization | Mathematical models | Composite structures | Convergence

Numerical evaluation of the order of convergence | Haar wavelet method | Extrapolation | Convergence theorem | Accuracy issues | NUMERICAL-SOLUTION | 2-DIMENSIONAL LEGENDRE WAVELETS | PARTIAL-DIFFERENTIAL-EQUATIONS | INTEGRAL-EQUATIONS | MATERIALS SCIENCE, COMPOSITES | SHELLS | FREE-VIBRATION ANALYSIS | Wavelet | Accuracy | Theorems | Discretization | Mathematical models | Composite structures | Convergence

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 06/2019, Volume 70, pp. 350 - 364

•Wavelet theory is a relatively new and emerging area in mathematical research.•Riemann–Liouville fractional integral operator for the Legendre wavelets has...

Legendre wavelets | Distributed order | Fractional differential equations | Caputo derivative | DELAY SYSTEMS | BLOCK-PULSE | CALCULUS | INTEGRODIFFERENTIAL EQUATIONS | MODEL | OPERATIONAL MATRIX | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | INTEGRATION | HYBRID | Algebra | Differential equations | Operators (mathematics) | Nonlinear equations | Numerical analysis | Numerical methods | Wavelet analysis | Formulas (mathematics)

Legendre wavelets | Distributed order | Fractional differential equations | Caputo derivative | DELAY SYSTEMS | BLOCK-PULSE | CALCULUS | INTEGRODIFFERENTIAL EQUATIONS | MODEL | OPERATIONAL MATRIX | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | INTEGRATION | HYBRID | Algebra | Differential equations | Operators (mathematics) | Nonlinear equations | Numerical analysis | Numerical methods | Wavelet analysis | Formulas (mathematics)

Journal Article

Journal of Vibration and Control, ISSN 1077-5463, 3/2018, Volume 24, Issue 6, pp. 1185 - 1201

This paper presents efficient numerical techniques for solving fractional optimal control problems (FOCP) based on orthonormal wavelets. These wavelets are...

DISSIPATION | fractional calculus | Fractional optimal control problem | Laguerre wavelets | CALCULUS | Legendre wavelets | EQUATIONS | Chebyshev wavelets | GENERAL FORMULATION | ENGINEERING, MECHANICAL | CAS wavelets | ACOUSTICS | OPERATIONAL MATRIX | MECHANICS | wavelet method | NUMERICAL-SOLUTIONS | Performance indices | Algebra | Lagrange multipliers | Optimal control | Differential equations | Chebyshev approximation | Wavelet analysis | Control theory | Trigonometric functions

DISSIPATION | fractional calculus | Fractional optimal control problem | Laguerre wavelets | CALCULUS | Legendre wavelets | EQUATIONS | Chebyshev wavelets | GENERAL FORMULATION | ENGINEERING, MECHANICAL | CAS wavelets | ACOUSTICS | OPERATIONAL MATRIX | MECHANICS | wavelet method | NUMERICAL-SOLUTIONS | Performance indices | Algebra | Lagrange multipliers | Optimal control | Differential equations | Chebyshev approximation | Wavelet analysis | Control theory | Trigonometric functions

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 2011, Volume 35, Issue 11, pp. 5235 - 5244

The solution of time-varying delay systems is obtained by using Chebyshev wavelets. The properties of the Chebyshev wavelets consisting of wavelets and...

Operational matrix | Chebyshev wavelets | Delay systems | Chebyshev polynomials | PARAMETER-ESTIMATION | INTEGRAL-EQUATIONS | LEGENDRE WAVELETS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | BLOCK-PULSE FUNCTIONS | VARIATIONAL-PROBLEMS | SINE-COSINE WAVELETS | LINEAR INTEGRODIFFERENTIAL EQUATION | 2ND KIND | FREDHOLM | Wavelet | Algebra | Mathematical analysis | Chebyshev approximation | Mathematical models | Matrices | Models | Delay

Operational matrix | Chebyshev wavelets | Delay systems | Chebyshev polynomials | PARAMETER-ESTIMATION | INTEGRAL-EQUATIONS | LEGENDRE WAVELETS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | BLOCK-PULSE FUNCTIONS | VARIATIONAL-PROBLEMS | SINE-COSINE WAVELETS | LINEAR INTEGRODIFFERENTIAL EQUATION | 2ND KIND | FREDHOLM | Wavelet | Algebra | Mathematical analysis | Chebyshev approximation | Mathematical models | Matrices | Models | Delay

Journal Article