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In this study, the stochastic distributed‐order time‐fractional version of the fourth‐order delay sub‐diffusion equation is defined by employing the Caputo fractional derivative. The orthonormal discrete Legendre polynomials, as a well‐known family of discrete polynomials basis functions, are used to develop a numerical method to solve this equation. To employ these polynomials in constructing the expressed approach, the operational matrices of the classical integration, differentiation (ordinary, fractional and distributed‐order fractional), and stochastic integration of these polynomials are extracted. The established method turns solving the introduced stochastic‐fractional equation into solving a more simple linear algebraic system of equations. In fact, by representing the unknown solution in terms of the introduced polynomials and employing the extracted matrices, this system is obtained. The accuracy of the developed algorithm is numerically checked by solving two examples. [ABSTRACT FROM AUTHOR]

This paper introduces a new set of the basis functions called the fractional Chebyshev cardinal wavelets and details their properties. These wavelets have a greater degree of freedom than the classical Chebyshev cardinal wavelets. Moreover, they retain the cardinality and the spectral accuracy of these wavelets. The fractional derivative and integral matrices of these fractional basis functions are obtained exactly in the explicit forms. In this study, we aim to devise an efficient and powerful approximation method using these new basis functions. Then, we employ them for a new category of nonlinear fractional quadratic integro-differential equations. By employing their fractional integral matrix and their cardinality, the problem under study is transformed into solving a nonlinear system of algebraic equations. The error analysis of the presented technique is first investigated theoretically and then computational efficiency is examined for two numerical examples. [ABSTRACT FROM AUTHOR]

In this study, a new family of cardinal wavelets called the extended Chebyshev cardinal wavelets is introduced to investigate problems on arbitrary finite domains. These wavelets possess many beneficial properties, such as spectral (exponential) accuracy, cardinality and orthogonality. The classical and fractional derivative matrices together with the fractional integral matrix of these wavelets are obtained. Two formulations of delay optimal control problems with a dynamical system involved with fractional derivatives are introduced. Two direct approaches based on these wavelets, together with their fractional derivative and integral matrices, are adopted for solving such problems. The presented methods transform solving the problems under study into solving constrained minimisation problems by approximating the state and control variables via the expressed wavelets. Some numerical examples are provided to show the efficiency of the expressed wavelet methods. [ABSTRACT FROM AUTHOR]

This study aims to generate a novel set of basis functions called the orthonormal piecewise Chelyshkov functions to solve a certain category of optimal control problems whose dynamical system is governed by a nonlinear fractional differential equation. A new fractional integral matrix associated with these basis functions is derived. This matrix significantly reduces the computations in solving such problems. The proposed approach transforms the original problem into a nonlinear programming one by expanding the control and state variables in terms of the orthonormal piecewise Chelyshkov functions and employing the derived fractional integral matrix. Some numerical problems are examined for verification of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the Chebyshev cardinal functions together with the extended Chebyshev cardinal wavelets are mutually utilized to generate a computational method for solving time fractional coupled Klein–Gordon–Schrödinger equations. By employing the cardinality of these basis functions, the nonlinear terms in the system under investigation are easily computed. This significantly reduces the difficulties and voluminous computational works. This approach does not need linearization, discretization and transformation techniques for obtaining a solution for the problem. The efficiency and accuracy of this approach are investigated in some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the one- and two-dimensional stochastic multi-order fractional diffusion-wave equations are introduced and a collocation procedure based on the shifted Jacobi polynomials is established to find their numerical solutions. Through this way, some operational matrices regarding classical and stochastic integrals as well as fractional and classical differentiations of these polynomials, are obtained. By representing the problem solution using an expansion of these polynomials (in which the coefficients of the expansion are unknown) and substituting it into the first problem, as well as by employing the obtained operational matrices, a system containing algebraic equations is obtained. Eventually, the coefficients of expansion and subsequently the solution of the original problem are found by solving this system. The correctness of the procedure is studied by solving four examples. [ABSTRACT FROM AUTHOR]

In this work, the Caputo fractional derivative defines the time fractional 3D Sobolev equation. The 2D shifted second kind Chebyshev cardinal polynomials (SSKCCPs) and 2D shifted second kind Chebyshev polynomials (SSKCPs) (as two well‐known classes of basis functions) are utilized to establish a hybrid technique for this new problem. First, the problem solution is approximated simultaneously using the 2D SSKCCPs (relative to x,y$$ x,y $$) and 2D SSKCCPs (relative to z,t$$ z,t $$). Next, the classical and fractional operational matrices of these polynomials are achieved and applied to make the hybrid algorithm. With a combination of derived operational matrices and the collocation approach, solving the expressed fractional 3D problem turns into solving an equivalent system of algebraic equations. The proposed algorithm's accuracy is checked using four numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical method for solving fractional optimal control problems by using hybrid functions is presented. The Riemann–Liouville fractional integral operator for hybrid functions is utilized to reduce the solution of optimal control problems to a nonlinear programming one, to which existing, well-developed algorithms may be applied. The method is computationally very attractive and gives very accurate results. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical method for solving the distributed fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann–Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

In this paper, a variable-order fractional version of the Benney–Lin equation is defined using the variable-order fractional derivative in the Caputo type. A collocation method based on the shifted Jacobi polynomials is applied to deal with this problem. Some matrix relationships related to these polynomials are extracted and used in constructing the established method. The obtained relations cause the method calculations to be significantly reduced, which reduce the method execution time. The established technique converts solving the problem under study into solving an algebraic system of equations. The high accuracy and low computations of the presented scheme are investigated by solving some numerical examples. [ABSTRACT FROM AUTHOR]

Background and Objective: Previous studies suggest a correlation of central venous pressure (CVP) and peripheral venous pressure (PVP) in different surgeries. CVP is one of essential monitoring during coronary arterial bypass graft (CABG). Canulation of peripheral venous catheter and PVP measurement is easier, whit less complication. This study was done to compare the central venous and peripheral venous pressures in the antecubital region in coronary arterial bypass graft surgery. Materials and Methods: This descriptive analytical study was done on 84 patients with American society of anesthesiology III with CABG surgery in Shafa hospital, Kerman, Iran during 2007. The technique of anesthesia was the same in all patients. CVP and PVP were measured every 20 minutes intervals before, on and after cardio-pulmonary pump. Data were analyzed using STAT-10, ANOVA and linear regression tests. Results: A total of 750 simultaneous measurements of CVP and PVP were recorded in all patients. The mean of CVP before of pump was 6.8±0.9 and the mean of PVP was 8.8±1. The mean difference between CVP and PVP was ±2 mmHg and this difference was significant (P<0.05). The mean of CVP on pump was 3.9±1 and the mean on PVP was 7.6±1. The mean difference of these two changeable was ±3.7 mmHg. This difference was significant (P<0.05). The mean of CVP after of pump was 6.5±1 and the mean of PVP was 8.5±1. The mean difference this two changeable was ±2 mmHg and this difference was significant (P<0.05). Conclusion: PVP and CVP are related with each other on cardio-pulmonary pump even in severe hemodynamic condition. PVP and CVP changes are synchronized and PVP is a helpful clinical alternative for estimation of CVP. [ABSTRACT FROM AUTHOR]

This paper examines a time fractional version of the 2D Fokker-Planck equation involved with the Atangana-Baleanu-Caputo fractional derivative, under the Dirichlet boundary conditions. A fully discretization approach based on the meshless local Petrov-Galerkin method and (3 − β) -order approximation is proposed for this equation. More precisely, we apply the meshless local Petrov-Galerkin method based on the Moving Kriging interpolation to discretize the space domain, and utilize the (3 − β) -order approximation together with the θ -weighted finite difference method to discretize the temporal domain. By implementing this method, we get a solution for the problem by solving a system of algebraic equations. The validity of the method is investigated by solving four examples. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a numerical method for solving nonlinear optimal control problems including state and control inequality constraints. The method is based upon rationalized Haar functions. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique. [Copyright &y& Elsevier]

This paper presents a computational technique for the solution of a class of nonlinear singular boundary value problems arising in physiology. The approach is based on a pseudospectral Legendre method in which Legendre-Gauss-Lobatto nodes are used to construct the Nth degree polynomial interpolation to approximate the solution. The method is easy to implement and yields accurate results. Illustrative examples are included to demonstrate the capability of the proposed method, and a comparison is made with existing methods in the literature. [ABSTRACT FROM AUTHOR]

Quartz crystal nanobalance (QCN) sensors are considered as powerful mass sensitive sensors to determine materials in the sub-nanogram level. In the current study a method based on QCN modified with polyethylene glycol (PEG) has been developed to determine organic vapors (ethyl acetate, acetone, ethanol and methyl ethyl ketone). The frequency shift of QCN was found to be linear against analytes concentrations in the range between 4 to 35 mg/L for acetone vapor and 4-70 mg/L for 3 other vapors. The correlation coefficients for ethyl acetate, acetone, ethanol, and methyl ethyl ketone were 0.9971, 0.9976, 0.9984 and 0.9927, respectively. The principal component analysis was also utilized to process the frequency response data of the organic vapors. Using principal component analysis, it was found that over 95% of the data variance could still be explained by use of two principal components (PC1 and PC2). Subsequently, the successful discrimination of ethyl acetate and other compounds was possible through the principal component analysis of the transient responses of the PEG-modified QCN sensor. [ABSTRACT FROM AUTHOR]

A numerical method for solving time-varying bilinear singular systems is presented. The method is based upon hybrid functions approximations. The properties of hybrid functions consisting of block-pulse functions and Legendre polynomials are presented. The operational matrices of integration and product are then utilized to reduce the solution of time-varying singular bilinear systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

Abstract: Rationalized Haar functions are developed to approximate the solution of the nonlinear Volterra–Fredholm–Hammerstein integral equations. The properties of rationalized Haar functions are first presented. These properties together with the Newton–Cotes nodes and Newton–Cotes integration method are then utilized to reduce the solution of Volterra–Fredholm–Hammerstein integral equations to the solution of algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples. [Copyright &y& Elsevier]

This article presents a numerical method for solving nonlinear mixed Volterra-Fredholm integral equations. The method is based on two-dimensional Legendre wavelet approximations. The properties of Legendre wavelets are utilized together with the Gaussian integration method to reduce the mixed integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

The Sinc-collocation method is presented for solving boundary value problems for nonlinear third-order differential equations. Properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce the computation of nonlinear third-order boundary value problems to some algebraic equations. The method is computationally attractive and applications are demonstrated through illustrative examples. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A numerical method for finding the solution of linear time-varying differential equations is proposed. The properties of the hybrid functions, which consist of block-pulse functions plus rationalized Haar functions, are presented. The properties of the hybrid functions together with the operational matrices of integration and product are then utilized to reduce the solution of differential equations to the solution of algebraic equations. Examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

Abstract: A method for finding the solution of linear time-varying multi-delay systems using a hybrid function is proposed. The properties of the hybrid functions which consist of block-pulse functions plus Taylor series are presented. The method is based upon expanding various time functions in the system as their truncated hybrid functions. Operational matrices of integration, delay and product are presented and are utilized to reduce the solution of multi-delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [Copyright &y& Elsevier]

In this paper we present a method for finding the solution of time-delay systems using a hybrid function. We present the properties of the hybrid functions, which consist of block-pulse functions plus Taylor series. The method is based upon expanding various time functions in the system as their truncated hybrid functions. Operational matrices of integration and delay are presented and are utilized to reduce the solution of time-delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

Abstract: The Sinc-Galerkin method is presented for solving nonlinear two-point boundary value problems for second order differential equations. A problem arising from chemical reactor theory is then considered. Properties of the Sinc-Galerkin method are utilized to reduce the computation of nonlinear two-point boundary value problems to some algebraic equations. The method is computationally attractive and applications are demonstrated through an illustrative example. [Copyright &y& Elsevier]

An approximate method for solving parabolic equations with a periodic boundary condition is proposed. The method is based upon using the Legendre series and the Hartley series to approximate the required solution. The parabolic equations are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. A numerical example is included to demonstrate the validity and applicability of the method and a comparison is made with existing results. [ABSTRACT FROM AUTHOR]

The Sine-collocation method is presented for solving Hallen's integral equations. Properties of Sinc functions are first presented, these properties are then utilized to reduce the computation of Hallen's integral equations to some linear algebraic equations. The method is computationally attractive, and applications are demonstrated through an illustrative example. [ABSTRACT FROM AUTHOR]

An approximate method for solving the diffusion equation with nonlocal boundary conditions is proposed. The method is based upon constructing the double shifted Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the diffusion equation with nonlocal boundary conditions are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results. [ABSTRACT FROM AUTHOR]

A direct method for solving variational problems using single-term Walsh series is presented. Two nonlinear examples are considered. In the first example the classical brachistochrone problem is examined, and in the second example a higher-order nonlinear problem is considered. The properties of single-term Walsh series are given and are utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement and yields accurate results. [ABSTRACT FROM AUTHOR]

This paper present a numerical method for solving the controlled Duffing oscillator. The method can be extended to nonlinear calculus of variations and optimal control problems. The method is based upon interpolating scaling functions. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

An approximate method for solving Volterra’s population model for population growth of a species in a closed system is proposed. Volterra’s model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. The approach is based on a rational Chebyshev tau method. The Volterra’s population model is first converted to a nonlinear ordinary differential equation. The operational matrices of derivative and product of rational Chebyshev functions are presented. These matrices together with the tau method are then utilized to reduce the solution of the Volterra’s model to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results. [Copyright &y& Elsevier]

A procedure is presented for solving the electric field integral equation for a thin wire. The method is based upon Alpert multiwavelets functions approximation. The properties of Alpert multiwavelets are first given. These wavelets are utilized to reduce the solution of sparse algebraic equations. In order to save memory requirement and computation time, a threshold procedure is applied to obtain sparse algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation. [ABSTRACT FROM AUTHOR]

An approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev (RC) tau method. The operational matrices of the derivative and product of RC functions are presented. These matrices together with the tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

A numerical method for solving the controlled Duffing oscillator is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions which consists of block-pulse functions plus Legendre polynomials are given. The method is based upon expanding various time functions in the system as their truncated hybrid functions. The operational matrix of product is introduced. This matrix together with the operational matrix of integration are utilized to reduce the solution of controlled Duffing oscillator to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [Copyright &y& Elsevier]

A numerical method for solving time varying linear quadratic optimal control problems with inequality constraints is presented. The method is based upon hybrid functions approximations. The properties of hybrid functions consisting of block-pulse functions and Legendre polynomials are presented. The operational matrices of integration and product are then utilized to reduce the optimal control problem to the solution of algebraic equations. The inequality constraints are first converted to a system of algebraic equalities. Illustrative examples are included to demonstrate the validity and applicability of the technique. [Copyright &y& Elsevier]

A method for finding the solution of time-varying singular bilinear systems using single-term Walsh series is proposed. The properties of single-term Walsh series are given and are utilized to find the solution of time-varying singular bilinear systems. [ABSTRACT FROM AUTHOR]

An approximate method for solving radiative Transfer equation in a slab medium with an isotropic scattering is presented. The method is based upon Legendre wavelets approximations. The differential and integral expressions which arise in the radiative transfer equation are converted into some linear systems of differential equations which can be solved for the unknown coefficient. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results. [ABSTRACT FROM AUTHOR]

An operational matrix of integration P based on sine and cosine wavelets is presented. A general procedure for forming this matrix is given. The matrix P is then used to reduce a variational problem to the solution of algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

In this paper rationalized Haar functions are developed to approximate the solutions of the nonlinear Fredholm-Hammerstein integral equations. Properties of rationalized Haar functions are first presented, and the operational matrix of the product of two rationalized Haar functions vector is utilized to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through an illustrative example. [ABSTRACT FROM AUTHOR]

An operational matrix of integration P based on Legendre wavelets is presented. A general procedure for forming this matrix is given. Illustrative examples are included to demonstrate the validity and applicability of the matrix P. [ABSTRACT FROM AUTHOR]

In this study, two categories of fractional optimal control problems are investigated. One category is optimal control problems, which includes a fractional dynamic system, and the other category, in addition to the fractional dynamics system, includes inequality constraints. The Atangana–Baleanu fractional derivative in the Caputo sense is used to define these problems. The extended Chebyshev cardinal wavelets as an appropriate class of basis functions are introduced to construct two numerical methods for these problems. To solve these problems, at first, an operational matrix for the Atangana–Baleanu fractional integral of these wavelets is derived. Then, by approximating the fractional derivative of the state variables and control variables in terms of the extended Chebyshev cardinal wavelets, and employing the fractional integral operational matrix of these wavelets and the Lagrange multipliers method, the problems under consideration are converted into systems of algebraic equations, which can be easily solved. To examine the accuracy of the established methods, some numerical examples are solved. [ABSTRACT FROM AUTHOR]

In this paper a one-dimensional inverse scattering problem is considered. An approximate inversion formula in a closed-form is derived. Using this formula, an approximate permitivity profile is generated from the partial knowledge of the frequency domain reflection coefficient. This formula is deduced through the application of a linear transformation to a nonlinear differential equation for the frequency domain reflection coefficient. Examples are given to demonstrate the accuracy of this approximate inversion formula. [ABSTRACT FROM AUTHOR]

Proposes a numerical method for solving the Gel'fand-Levitan integral equation and an alternative method for the construction of smooth dielectric profiles. Two examples given to demonstrate the accuracy of the proposed method.