Existence and Uniqueness of Nonlinear Volterra Integral Equations With Variable Fractional Order in Fréchet Spaces via a Frigon−Granas Fixed Point Approach

In this paper, we proved the existence and uniqueness and convergence of the solution of new type for nonlinear fuzzy volterra integral equation . The homotopy analysis method are proposed to solve the new type fuzzy nonlinear Volterra integral equation . We convert a fuzzy volterra integral equation for new type of kernel for integral equation, to a system of crisp function nonlinear volterra integral equation . We use the homotopy analysis method to find the approximate solution of the system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy volterra integral equation . Some numerical examples is given and results reveal that homotopy analysis method is very effective and compared with the exact solution and calculate the absolute error between the exact and AHM .Finally using the MAPLE program to solve our problem .

In this research, we present novel generalized derivative founded on the newly constructed (NGCFVODs) novel generalized Caputo Fractional variable order derivatives. Utilizing these operators, a numerical methodology has been formulated to resolve the Fractional Variable Order Differential Equations (FVODEs). We estimate the solution for (FVODEs) by employing bernstein polynomials as foundational vectors. We have further expanded the derivative operational matrix of bernstein Polynomials (bPs) to generalized derivative an operational matrix in sense of (NGCFVODs). The efficiency of developed numerical methodology is tested by a taking a various test examples. We also a compare results of our suggested approach with the methodologies existing in academic papers. In this study the Fractional variable order differential operator of new generalized Caputo was described by three categories: (i) various value in ρ and Fractional variable order a parameter, (ii) various value in a fractional parameters while Fractional variable order and ρ parameter are fixed, and (iii) various value in Fractional variable order parameter controlling fractional and ρ parameter.

Common solution to a pair of nonlinear Fredholm and Volterra integral equations and nonlinear fractional differential equations

Constant fractional order vibration control strategy has been one of research hotspots in recent decades. However, the variable fractional order control method is seldom concerned up to now. In this article, a novel variable fractional order sliding mode control (VOSMC) method is proposed to suppress the responses of building structure caused by seismic excitations, including El Centro, Hachinohe, Northridge, and Kobe earthquakes. Based on the proposed variable fractional order sliding mode surface, the control law of VOSMC is presented. The global asymptotic stability of the control system is analyzed and proved by utilizing variable fractional order Lyapunov stability theorem. Besides, the corresponding constant fractional order sliding mode control (COSMC) method is also given. The control effects of VOSMC and COSMC methods are discussed by four performance indices. Finally, the utilizability and reasonability of the proposed control method is verified by using two examples (include two-story and five-story shear buildings). Compared with the COSMC method, the proposed variable fractional order controller not only has a lesser control output, but also has a higher utilization of the output, which is conducive to energy saving.

In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference.

The variable fractional dimensions differential and integral operator overrides the phenomenon of the constant fractional order. This leads to exploring some new ideas in the proposed direction due to its varied applications in the recent era of science and engineering. The present papers deal with the replacement of the constant fractional order by variable fractional order in various fractal-fractional differential equations. An advanced numerical scheme is developed with the help of Lagrange three-point interpolation and further, it is employed for the solution of the proposed differential equations. However, the properties of these new operators are presented in detail. Finally, the error analysis is also conducted for the numerical scheme deployed. The results are validated by the suitable choice of applications to real-life problems. The well- known multi-step-Adams–Bashforth numerical scheme for classical differential equations is recovered when the non-integer order is one.

The strain paths of cement stone in the deviatoric and meridian planes under the constant Lode angle loading path (true triaxial stress state) are analyzed. The amount of volumetric and shear strains first increases and then decreases with the intermediate principal stress coefficient. Owing to the generation of plastic volumetric strain and plastic shear strain in the direction of deviatoric stress, the strain path exhibits nonlinearity in the meridian planes. The deviation of the strain path from the constant Lode angle arises from the accumulation of plastic shear strain along the Lode angle direction. In the framework of fractional plasticity, a three-dimensional elastoplastic constitutive model incorporating Lode angle is proposed, including yield function, potential function, and fractional flow rule. The yield surface evolves in both meridian and deviatoric planes, allowing the yield function to precisely characterize the stress state. Since the plus-minus sign in the flow direction of the yield surface is opposite to that in the flow direction of cement stone, a simple elliptic function incorporating Lode angle serves as the potential function. The procedure for the determination of fractional order based on the entirety of the deformation process is proposed, including variable and constant fractional order. The comparison between the experimental result and the analytical solution of constitutive model confirms its accuracy and validity. Furthermore, the difference between variable and constant fractional order on deformation is analyzed. The comparison results indicate that the variable fractional order can provide a more accurate description of deformation than the constant fractional order.

A modified numerical technique was developed to solve a wide class of variable order fractional optimal control problems in the sense of Riemann Liouville or Caputo derivatives. The modified algorithm is based on the non-standard finite difference method of solving fractional differential equations of variable order. Important property of a reflection operator is used to simplify the variable order right Riemann Liouville or Caputo derivatives to the variable order left Riemann Liouville or Caputo derivatives. Necessary and sufficient conditions that guarantee the existence and the uniqueness of the solution of the resulting systems are given. Illustrative examples are included to demonstrate the validity and the effectiveness of the established approach.

Shifted Legendre Polynomials algorithm used for the dynamic analysis of viscoelastic pipes conveying fluid with variable fractional order model

In this study, new generalized derivative and integral operators are introduced, stemming from the newly developed new generalized Caputo variable order fractional derivatives (NGCVFDs). Utilizing these operators, a numerical method is devised to address variable order fractional differential equations (VOFDEs). The solutions of VOFDEs are approximated using shifted Legendre polynomials (SLPs) as basis vectors, and the derivative operational matrix of SLPs is extended to a generalized derivative operational matrix in the context of NGCVFDs. The efficiency of the numerical method is assessed through various test examples. Additionally, the outcomes of the proposed method are compared with existing methodologies in the literature. The variable-order fractional differential operators of the generalized Caputo is categorized into three types in this paper: (i) Different values in ρ and Fractional variable order parameters, (ii) Different values in fractional parameter whilst ρ parameters and Fractional variables orders are constant, and (iii) Different values in Fractional variables orders parameters controlled fraction and ρ parameters. The example of numerical methods show theoretical interpretation and prove effectiveness of suggested technique.

In this study, we explore the epidemic spread of the coronavirus using the Caputo fractional variable order derivative as variable order derivative provides a natural extension to classical as well as fractional order derivatives. Using the variable order derivatives in investigation of biological models of infectious diseases is an important area of research in the current time. Using the fixed point technique, we discuss the existence and uniqueness of solution to the corona virus infectious disease 2019 environmental transformation model. In order to demonstrate the existence and novelty of our findings, we examine the results numerically and graphically with the help of Euler’s method. There are several graphs provided that are related to different variable orders.

Homotopy perturbation method for solving time-fractional nonlinear Variable-Order Delay Partial Differential Equations

A spectral iterative method for solving nonlinear singular Volterra integral equations of Abel type

Variational fractional-order modeling of viscoelastic axially moving plates and vibration simulation

Variable fractional order Riccati equation is solved to get the exact solution by mixing Sumudu transform and Adomain decomposition method. This procedure is powerful and takes short computations’ as in the illustrative examples.