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In the current research, we develop a collocation method based on the biorthogonal Hermite cubic spline functions to solve a class of fractional optimal control problems using Caputo–Fabrizio derivative operator. We design dual bases for Hermite cubic spline functions for the first time in this work. So, we present two direct and efficient algorithms to solve a class of fractional optimal control problems. Two new operational matrices of Caputo–Fabrizio fractional derivative are derived using Hermite cubic spline functions. By employing these matrices, we reduce these problems to systems of algebraic equations. Illustrative examples are examined to demonstrate the important features of the new algorithm.

Background: Medical ultrasonic images are usually degraded by a special kind of noise called ‘speckle’. The speckle noises usually have an effect more on edges and fine details of an ultrasound images which lead to reduction in their contrast resolution consequently create difficulties in the diagnosis of illnesses. Methods: In this paper, to reduce the speckle noise of medical ultrasound image, we have proposes a denoising approach by using ‘discrete shearlet transform’ based on pseudo-polar fast Fourier transform. The mathematical structure of present method is briefly discussed. To illustrate the efficiency of proposed approach, the experimental results of synthesis of some real Ultrasound images are considered. Results: Experimental results illustrate that proposed approach can obtain better performance in terms of Peak Signal To Noise Ratio (PSNR) for ultrasound image denoising. Our numerical tests include a series of computer-simulated and in MATLAB experiments.

In this paper, a TV-based shearlet shrinkage is proposed for discontinuity-preserving denoising using a combination of shearlet with a total variation scheme. For TV denoising numerical procedure, we use two approaches. In the first approach, we apply semi-implicit method for total variation. To solve Euler–Lagrange equation associated with total variation, we use additive operator splitting (AOS) scheme. This approach has good effect on suppressing the pseudo-Gibbs and shearlet-like artifacts and is very efficient for reducing iterations. In the second approach, we use Split Bregman iteration method. This techniques converges very quickly and in combine by shearlet shrinkage produce good results.

The aim of present paper is to obtain the analytical solution of modified Fornberg–Whitham equation by using $$\tan (\phi /2)$$ tan(ϕ/2) -expansion and $$\tanh (\phi /2)$$ tanh(ϕ/2) -expansion methods. These methods are used to construct solitary and soliton solutions of nonlinear evolution equation. These methods are straightforward and concise, and their applications are promising. These methods are developed for searching exact traveling wave solutions of nonlinear partial differential equations. The exact particular solutions containing five types hyperbolic function solution, trigonometric function solution, exponential solution, logarithmic solution and rational solution. The expansion methods presents a wider applicability for handling nonlinear wave equations. Also, their are shown that the expansion methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear evolution equations.

A numerical technique based on the finite difference and collocation methods is presented for the solution of Korteweg–de Vries (KdV) equation. The integration relations between any two families of B-spline functions are presented and are utilized to reduce the solution of KdV equation to the solution of linear algebraic equations. Numerical simulations for three test examples have been demonstrated to validate the technique proposed in the current paper. It is found that the simulating results are in good agreement with the exact solutions.

In this paper, we applied the improved $$\tan \left( \Phi (\xi )/2\right) $$tanΦ(ξ)/2-expansion scheme for the (2+1)-dimensional Zoomeron, the Duffing and the symmetric regularized long wave equations and exact particular solutions have been found. The exact particular solutions containing four types hyperbolic function solution, trigonometric function solution, exponential solution and rational solution. We obtained the further solutions comparing with other methods as sine–cosine function method (Qawasmeh in J Math Comput Sci 3:1475–1480, 2013). Recently this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that this method, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.

The main idea of this study is to introduce a novel set of wavelet functions named Touchard wavelets for solving time-fractional Black–Scholes equations. The numerical method is discussed based on the pseudo-operational matrix of Riemann–Liouville fractional integration and the least square approximation method. The methodology of deriving the pseudo-operational matrix is calculated accurately and this has a direct effect on the accuracy of the method. The error estimation is proposed. At last, two numerical experiments are employed to clarify the performance and high accuracy of the present technique.

Abstract: Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. Here we construct the operational matrix of fractional derivative of order α in the Caputo sense using the linear B-spline functions. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus we can solve directly the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the new technique presented in the current paper. [Copyright &y& Elsevier]

The main aim of this paper is to apply the trigonometric wavelets for the solution of the Fredholm integro‐differential equations of nth‐order. The operational matrices of derivative for trigonometric scaling functions and wavelets are presented and are utilized to reduce the solution of the Fredholm integro‐differential equations to the solution of algebraic equations. Furthermore, we get an estimation of error bound for this method. Copyright © 2011 John Wiley & Sons, Ltd.

Abstract: Two numerical techniques based on the finite difference and collocation methods are presented for the solution of nonlinear Klein–Gordon equation. The operational matrix of derivative for the cubic B-spline scaling functions is presented and is utilized to reduce the solution of nonlinear Klein–Gordon equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new techniques. [Copyright &y& Elsevier]

Abstract: Two numerical techniques are presented for solving the solution of Riccati differential equation. These methods use the cubic B-spline scaling functions and Chebyshev cardinal functions. The methods consist of expanding the required approximate solution as the elements of cubic B-spline scaling function or Chebyshev cardinal functions. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the new techniques. The methods are easy to implement and produce very accurate results. [Copyright &y& Elsevier]

A numerical technique is presented for the solution of the second order onedimensional linear hyperbolic equation. This method uses the Chebyshev cardinal functions. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, the problem is reduced to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

In this article a numerical technique is presented for the solution of Fokker‐Planck equation. This method uses the cubic B‐spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B‐spline scaling function. Using the operational matrix of derivative, the problem will be reduced to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Problems for parabolic partial differential equations with nonlocal boundary conditions have been studied in many articles, but boundary value problems for hyperbolic partial differential equations have so far remained nearly uninvestigated. In this article a numerical technique is presented for the solution of a nonclassical problem for the one‐dimensional wave equation. This method uses the cubic B‐spline scaling functions. Some numerical results are reported to support our study. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

In this paper, we have presented an accurate and impressive spectral algorithm for solving fractional telegraph equation with Riesz-space derivative and Dirichlet boundary conditions. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrices of Riemann–Liouville fractional integral and left- and right-sided Caputo fractional derivatives. Primarily, we implement the proposed algorithm in both temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations which considerably simplifies the problem. In addition, an error bound is established in the L8-norm for the suggested spectral Jacobi tau method. Illustrative examples are included to demonstrate the validity and accuracy of the presented technique.