An Approach to Solve Fuzzy Fractional Darboux Problems Under the Caputo Derivative

The aims of this paper are to propose mixed fractional derivative, clarify the problems suggested by the tacit approach of finite difference, and the study of consistency, stability, and convergence methods. An effective computational approach for solving Factional Percolation Equations Riesz Space Mixed Fractional Derivative (FPERSMFD) using Implicit Finite Difference Methods (IFDMs) is proposed. The moved Grunwald estimate is analyzed for the mixed fractional derivatives. However, the given method is successfully applied to the mixed fractional derivative classes with Riesz space in order to solve various Fractional Percolation Equations (FPEs). The numerical method of fractional order is defined as consistent, stable, and convergent. Four illustrative examples are given to illustrate the efficiency and the validity of the algorithm proposed and to compare the results with the exact solution. For these four examples and from the Tables illustrated in this work, a high-precision approximation, the approximate solution values of the various grid points provided by the implicit finite difference methods are similar to the exact solution values. Different and random values for fractional derivative have been used to prove the efficiency of this proposed method where the error equals to zero between the proposed method and the exact method. It can also be seen that the accuracy improves with the approximation order. The data have been presented in Tables by using the software package MathCAD 12 and MATLAB when necessary. In solving Factional percolation equations Riesz space mixed fractional derivative, the implicit finite difference methods have appeared to be effective and reliable.

We study mixed Riemann-Liouville fractional integration operators and mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The obtained are results generalized to the case of Hölder spaces with power weight. Keywords: functions of two variables, fractional derivative of Marchaud form, mixed fractional derivative, weight, mixed fractional integral, Hölder space.

This study develops a new definition of a fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. This developed definition encompasses many types of fractional derivatives, such as the Riemann–Liouville and Caputo fractional derivatives for singular kernel types, as well as the Caputo–Fabrizio, the Atangana–Baleanu, and the generalized Hattaf fractional derivatives for non-singular kernel types. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, a novel numerical scheme is developed to approximate the solutions of a class of fractional differential equations (FDEs) involving the mixed fractional derivative. Finally, an application in computational biology is presented.

The aims of this paper are to propose approach of explicit finite difference mathod (EFDM), clarify the problem the mixed fractional derivative in one-dimensional fractional percolation equation (O-DFPE), and the study of consistency, stability, and convergence methods. Use of estimated Grunwald estimation in the analysis of mixed fractional derivatives. However, the given method is successfully applied to the mixed fractional derivative classes with the initial condition (IC) and derivative boundary conditions (DBC). To illustrate the efficiency and validity of the proposed algorithm, examples are given and the results are compared with the exact solution. From the figures shown for the examples in this work, the approximate solution values given by the EFDM for the various grid points are equivalent to the exact solution values with high-precision approximation. To show the effectiveness of the proposed method, where the error between the EFDM and the exact method is zero, the fractional derivative was used with various and random values. Using the package MATLAB and MathCAD 12 Figures were introduced.

We consider two types of partial fractional differential equations in two dimensions with mixed fractional derivatives. Appropriate Lyapunov-type inequalities are proved, and applications to the certain eigenvalue problems are presented. Moreover, some connections with the fractional variational problems are highlighted.

The optimization of shift‐and‐add network for constant multiplications is found to have great potential for reducing the area, delay, and power consumption of implementation of multiplications in several computation‐intensive applications not only in dedicated hardware but also in programmable computing systems. To simplify the shift‐and‐add network in single constant multiplication (SCM) circuits, this chapter discusses three design approaches, including direct simplification from a given number representation, simplification by redundant signed digit (SD) representation, and simplification by adder graph. Examples of the multiple constant multiplication (MCM) methods are constant matrix multiplication, discrete cosine transform (DCT) or fast Fourier transform (FFT), and polyphase finite impulse response (FIR) filters and filter banks. The given constant multiplication methods can be used for matrix multiplications and inner‐product; and can be applied easily to image/video processing and graphics applications. The chapter further discusses some of the shortcomings in the current research on constant multiplications, and possible scopes of improvement.

A reliable method for the numerical solution of the kinetics problems

In this research paper, we applied the Adomian's decomposition method to determine the analytical exact solutions of linear and nonlinear Goursat problems which play very important part in applied and engineering sciences. The proposed technique is fully compatible with the complexity of these problems and obtained results are highly encouraging. Some examples with closed form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient. presence of small parameters in the differential equation, and provides the solution (or an approximation to it) as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives. The Adomian's Decomposition Method has been shown to solve effectively, easily, and accurately a large class of linear and nonlinear problems, generally two or three iterations lead to high accurate solutions. The basic motivation of the present study is to extend the application of Adomian's decomposition method to linear and nonlinear Goursat problems.

In this paper, we are going to study a nonlinear fractional partial differential equations by double Sumudu transformation coupled with Adomian decomposition method or with variational iteration method. For this case, we choose an important fractional partial differential equations, such as parabolic-hyperbolic with mixed fractional derivatives (homogeneous and nonhomogeneous) types. Some examples are given here to illustrate efficiency of this method.

The method of lower and upper solutions for mixed fractional four-point boundary value problem with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>p</mml:mi></mml:math>-Laplacian operator

In this paper, we investigate the existence of positive solutions for the boundary value problem of nonlinear fractional differential equation with mixed fractional derivatives and p-Laplacian operator. Then we establish two smart generalizations of Lyapunov-type inequalities. Some applications are given to demonstrate the effectiveness of the new results.

We obtain sufficient conditions for the existence and uniqueness of a solution of a boundary value problem for a differential equation that contains a mixed fractional derivative.

Purpose This paper aims to discuss the approximate solution of the nonlinear thin film flow problems. A new analytic approximate technique for addressing nonlinear problems, namely, the optimal homotopy asymptotic method (OHAM), is proposed and used in an application to the nonlinear thin film flow problems. Design/methodology/approach This approach does not depend upon any small/large parameters. This method provides a convenient way to control the convergence of approximation series and to adjust convergence regions when necessary. Findings The obtained solutions show that the OHAM is more effective, simpler and easier than other methods. The results reveal that the method is explicit. By applying the method to nonlinear thin film flow problems, it was found to be simpler in applicability, and more convenient to control convergence. Therefore, the method shows its validity and great potential for the solution of nonlinear problems in science and engineering. Originality/value The proposed method is tested upon nonlinear thin film flow equation from the literature and the results are compared with the available approximate solutions including Adomian decomposition method (ADM), homotopy perturbation method, modified homotopy perturbation method and HAM. Moreover, the exact solution is compared with the available numerical solutions. The graphical representation of the solution is given by Maple and is physically interpreted.

Integral methods -such as the Finite Element Method (FEM) and the Boundary Element Method (BEL1)are frequently used in structural optimization problems to solve systems of partial differential equations. Therefore: one must take into account the large computational requirements of these sophisticated techniques at the time of choosing a suitable Mathematical Programming (MP) algorithm for this kind of problems. Among the currently available M P algorithms, Sequential Linear Programming (SLP) seems t o be one of the most adequate to structural optimization. Basically, SLP consist in constructing succesive linear approximations to the original non linear optimization problem within each step. However, the application of SLP may involve important malfunctions. Thus, the solution to the approximated linear problems can fail to exist, or may lead to a highly unfeasible point of the original non linear problem; also, large oscillations often occur near the optimum, precluding the algorithm to converge. In this paper, we present an improved SLP algorithm with line-search. specially designed for structural optimization problems. In each iteration) an approximated linear problem with aditional side constraints is solved by Linear Programming. The solution to this linear problem defines a search direction. Then, the objective function and the non linear constraints are quadratically approximated in the search direction, and a line-search is performed. The algorithm includes strategies t o avoid stalling in the boundary of the feasible region, and to obtain alternate search directions in the case of incompatible linearized constraints. Techniques developed by the authors for efficient high-order shape sensitivity analysis are referenced. Transactions on the Built Environment vol 52, © 2001 WIT Press, www.witpress.com, ISSN 1743-3509

A nontraditional approach to the nonlinear inverse boundary value problem is illustrated using multiple examples of the Poisson equation. The solutions belong to a class of analytical solutions defined through Bézier functions. The solution represents a smooth function of high order over the domain. The same procedure can be applied to both the forward and the inverse problem. The solution is obtained as a local minimum of the residuals of the differential equations over many points in the domain. The Dirichlet and Neumann boundary conditions can be incorporated directly into the function definition. The primary disadvantage of the process is that it generates continuous solution even if continuity and smoothness are not expected for the solution. In this case they will generate an approximate analytical solution to either the forward or the inverse problem. On the other hand, the method does not need transformation or regularization, and is simple to apply. The solution is also good at damping the perturbations in measured data driving the inverse problem. In this paper we show that the method is quite robust for linear and nonlinear inverse boundary value problem. We compare the results with a solution to a nonlinear inverse boundary value problem obtained using a traditional approach. The application involves a mixture of symbolic and numeric computations and uses a standard unconstrained numerical optimizer.