Class Of Initial Value Problems Research Articles

A numerical method for solutions of tempered fractional differential equations

The main objective of this paper is to develop a technique for approximating the numerical solution of certain class of tempered fractional differential equations. The proposed technique is based on the Newton–Cotes quadrature formula, and it is established by using generalized Lagrange interpolation polynomials. The product integration technique is used to develop a formula for the left and right-sided tempered fractional integrals. Next, by using these formulas, a method is developed for solving a general class of tempered fractional differential equations. Furthermore, three distinct classes of initial value problems and one of the terminal value problems for tempered fractional differential equations are taken into consideration and each class is illustrated by a numerical experiment to demonstrate the validity and effectiveness of the proposed methodology. Moreover, upper bound of error in the numerical approximations have been provided.

Fourier-Gegenbauer pseudospectral method for solving time-dependent one-dimensional fractional partial differential equations with variable coefficients and periodic solutions

In this paper, we present a novel pseudospectral (PS) method for solving a new class of initial-value problems (IVPs) of time-dependent one-dimensional fractional partial differential equations (FPDEs) with variable coefficients and periodic solutions. A main ingredient of our work is the use of the recently developed periodic RL/Caputo fractional derivative (FD) operators with sliding positive fixed memory length of Bourafaet al. (2021) or their reduced forms obtained by Elgindy (2023) as the natural FD operators to accurately model FPDEs with periodic solutions. The proposed method converts the IVP into a well-conditioned linear system of equations using the PS method based on Fourier collocations and Gegenbauer quadratures. The reduced linear system has a simple special structure and can be solved accurately and rapidly by using standard linear system solvers. A rigorous study of the computational storage requirements as well as the error and convergence of the proposed method is presented. The idea and results presented in this paper are expected to be useful in the future to address more general problems involving FPDEs with periodic solutions.

Incommensurate non-homogeneous system of fuzzy linear fractional differential equations using the fuzzy bunch of real functions

This article aims to introduce and investigate the analytical fuzzy solution of the incommensurate non-homogeneous system of fuzzy linear fractional differential equations (INS-FLFDEs) using trivariate Mittag-Leffler functions. Entries of the coefficient matrix of the given system are treated as real numbers, initial-values are triangular fuzzy numbers (TFNs), and the forcing function is a fuzzy set (or a bunch) of real function. We extract the potential solution in the form of a fuzzy bunch of real functions (FBoRFs) rather than the solution of fuzzy-valued functions. We formulate the fuzzy initial value problem as a set of classical initial value problems by taking the forcing function from the class of FBoRFs and the initial value from the collection of TFNs (as a special case). The solution of this system is in the form of a trivariate Mittag-Leffler function. We interpret this solution as an element of the fuzzy solution set and assign the minimum value of membership that takes from the forcing function and the initial value in the fuzzy set. The originality of the proposed technique is that the uncertainty is smaller compared to the uncertainty extracted from other techniques. In addition, generalized derivatives increase the order and dimension of the system. Therefore, the proposed technique is better in terms of complexity because it reduces the order and dimension of the system. Finally, to grasp the proposed technique, we solve the electrical network and multiple mass-spring systems as applications and analyze their graphs to visualize and support theoretical results.

A new variational discretization technique for initial value problems bypassing governing equations

Motivated by the fact that both the classical and quantum description of nature rest on causality and a variational principle, we develop a novel and highly versatile discretization prescription for classical initial value problems (IVPs). It is based on an optimization (action) functional with doubled degrees of freedom, which is discretized using a single regularized summation-by-parts (SBP) operator. Formulated as optimization task it allows us to obtain classical trajectories without the need to derive an equation of motion. The novel regularization we develop in this context is inspired by the weak imposition of initial data, often deployed in the modern treatment of IVPs and is implemented using affine coordinates. We demonstrate numerically the stability, accuracy and convergence properties of our approach in systems with classical equations of motion featuring both first and second order derivatives in time.

Existence of solutions for impulsive wave equations

<abstract><p>We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. To prove our main results we give a suitable integral representation of the solutions of the considered problem. Then, we construct two operators so that any fixed point of their sum is a solution.</p></abstract>

Impulsive fractional differential equations with state-dependent delay involving the Caputo-Hadamard derivative

In this paper, we investigate the existence of solutions for a class of initial value problems for impulsive Caputo-Hadamard fractional differential equations with state-dependent delay.

Two nonnegative solutions for two-dimensional nonlinear wave equations

We study a class of initial value problems for two-dimensional nonlinear wave equations. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions. The arguments are based upon a recent theoretical result.

Effective Optimized Decomposition Algorithms for Solving Nonlinear Fractional Differential Equations

Abstract In this paper, the optimized decomposition method, which was developed to solve integer-order differential equations, will be modified and extended to handle nonlinear fractional differential equations. Fractional derivatives will be considered in terms of Caputo sense. The suggested modifications design new optimized decompositions for the series solutions depending on linear approximations of the nonlinear equations. Two optimized decomposition algorithms have been introduced to obtain approximate solutions of broad classes of initial value problems (IVPs) consisting of nonlinear fractional ordinary differential equations (ODEs) and partial differential equations (PDEs). A comparative study was conducted between the proposed algorithms and the Adomian decomposition method (ADM) by means of some test illustration problems. The implemented numerical simulation results showed that the proposed algorithms give better accuracy and convergence, and reduce the complexity of computational work compared to the Adomian's approach. This confirms the belief that the optimized decomposition method will be used effectively and widely as a powerful tool in solving various fractional differential equations.

Analysis of New Type of Second-order Fractional Linear Multi-step Method with Improved Stability

We present and investigate a new type of implicit fractional linear multi-step method of order two for fractional initial value problems. The method is obtained from the second-order superconvergence of the Grünwald-Letnikov approximation of the fractional derivative at a non-integer shift point. The method coincides with the backward difference method of order two for the classical initial value problem when the order of the derivative is one. The weight coefficients of the proposed method are obtained from the Grünwald weights and are hence computationally efficient compared with that of the fractional backward difference formula of order two. The stability properties are analyzed and it is shown that the stability region of the method is larger than that of the fractional Adams-Moulton method of order two and the fractional trapezoidal method. Numerical results and illustrations are presented to justify the analytical theories.

Uniqueness of Local, Analytic Solutions to Singular ODEs

We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem to an equilibrium problem of a regular ODE. Then, we apply classical invariant manifold theory. We demonstrate that the class of ODEs under consideration captures models which describe the shape of axially symmetric surfaces which are closed on one side. Our main result guarantees smoothness at the tip of the surface.

Well-posedness and stability for fuzzy fractional differential equations

In this article, we consider the existence and uniqueness of solutions for a class of initial value problems of fuzzy Caputo–Katugampola fractional differential equations and the stability of the corresponding fuzzy fractional differential equations. The discussions are based on the hyperbolic function, the Banach fixed point theorem and an inequality property. Two examples are given to illustrate the feasibility of our theoretical results.

On the solution of Volterra integral equations with decomposable kernel functions

We consider the solution of a class of nonlinear Volterra integral equations, x(t)=g(t)+∫⁡t0tk(t,s;x(s))ds, where t≥t0 and the kernel function k is finitely decomposable, and derive variation of parameters formulae that provide the solution of corresponding perturbed nonlinear equations. We attain this by relating the integral equations to a certain class of initial value problems for ODEs, for which variation of parameters are also formulated.

Existence and ulam stability of k-generalized ψ-Hilfer fractional problem

In this paper, we prove existence, uniqueness stability results for a class of initial value problem for fractional differential equations involving generalized ψ-Hilfer fractional derivative. The result is based on the Banach contraction mapping principle. In addition, two examples are given to illustrate our results.

Implicit fractional differential equations with Φ–Caputo fractional derivative in Banach spaces

In the present paper, we consider a class of initial value problem for nonlinear implicit fractional differential equations involving the Φ–Caputo fractional derivative in Banach spaces. A new minimal condition on the parameters and the relationship between them is used to show the existence, uniqueness, and Mittag-Leffler–Ulam–Hyers stability of solutions by using the Mönch’s fixed point theorem combined with a classical technique for measures of noncompactness. Moreover, the Mittag-Leffler–Ulam–Hyers stability of proposed problem is discussed. Finally, two examples are presented to illustrate the results.

Neutral functional sequential differential equations with Caputo fractional derivative on time scales

In this paper, we establish the existence and uniqueness of a solution for a class of initial value problems for implicit fractional differential equations with Caputo fractional derivative. The arguments are based upon the Banach contraction principle, the nonlinear alternative of Leray–Schauder type and Krasnoselskii fixed point theorem. As applications, two examples are included to show the applicability of our results.

On positivity and monotonicity analysis for discrete fractional operators with discrete Mittag–Leffler kernel

We consider conditions under which the positivity of a fractional difference implies either positivity, monotonicity, or convexity, and we consider both the non‐sequential and sequential cases. The particular difference we study is the discrete nabla ABC‐type difference, and it possesses a Mittag–Leffler kernel; this leads to some different results when compared to other types of kernels. We conclude by applying our results to certain classes of initial value problems in discrete fractional calculus.

Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo gH-Type Derivatives

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo gH-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of gH-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and ε-approximate solution of the coupled system. Finally, we propose some work for future research.

On a nonlocal implicit problem under Atangana\u2013Baleanu\u2013Caputo fractional derivative

In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder’s and Banach’s fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.

Existence and uniqueness of solutions to impulsive fractional differential equations via the deformable derivative

In this paper, we establish sufficient conditions for the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the deformable fractional derivative. We achieve our results using classical fixed point theorems such as the Banach contraction principle, the Shaefer and the alternative Leray–Schauder theorems. We provide an example to illustrate our abstract results.

Bayesian numerical methods for nonlinear partial differential equations

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of PDEs is identified using novel theoretical analysis of the sample path properties of Matérn processes, which may be of independent interest.