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We use power series with rational exponents to find exact solutions to initial value problems for fractional differential equations. Certain problems that have been previously studied in the literature can be solved in a closed form, and approximate solutions are derived by constructing recursions for the relevant expansion coefficients.

The work is devoted to developing the parallel algorithms for solving the initial boundary problem for the time-fractional diffusion equation. After applying the finite-difference scheme to approximate the basis equation, the problem is reduced to solving a system of linear algebraic equations for each subsequent time level. The developed parallel algorithms are based on the Thomas algorithm, parallel sweep algorithm, and accelerated over-relaxation method for solving this system. Stability of the approximation scheme is established. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to compare these methods and to study the performance of parallel implementations. The parallel sweep method shows the lowest computing time. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results.

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results.

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results.

In recent years, fractional differential equations have been extensively applied to model various complex dynamic systems. The studies on highly accurate and efficient numerical methods for fractional differential equations have become necessary. In this paper, an effective recurrence algorithm for computing both the fractional Riemann-Liouville and Caputo derivatives is proposed, and then spectral collocation methods based on the algorithm are investigated for solving fractional differential equations. By the recurrence method, the numerical stability with respect to N, the number of collocation points, can be improved remarkably in comparison with direct algorithm. Its robustness ensures that a highly accurate spectral collocation method can be applied widely to various fractional differential equations.

A Master of Science thesis in Mathematics by Razan Abdul Satar Alchikh entitled, “Solution of Fractional Differential Equations: Variational Iteration Approachs”, submitted in December 2019. Thesis advisor is Dr. Suheil Khoury. Soft copy is available (Thesis, Approval Signatures, Completion Certificate, and AUS Archives Consent Form)., Fractional calculus is a novel and highly active area of research in the literature as it has a wide spectrum of applications in the physical sciences and engineering. In this thesis, we study the numerical solution of fractional differential equations subject to initial or boundary conditions. We employ two iterative methods for the solution of such equations. First, we implement a Laplace decomposition method (LDM) which is a combination of two methods: Laplace transform and a decomposition scheme. The nonlinear term is decomposed and a recursive algorithm is composed for the determination of the proposed infinite series solution. Second, we present another iterative technique that is based on the incorporation of Green’s functions into well-established fixed-point iterations, including Krasnoselskii-Mann’s and Picard’s schemes. Numerical experiments are conducted to demonstrate the efficiency, accuracy and applicability of the proposed methods, and then, to compare them with other schemes. In addition, we present graphs to understand the behavior of the numerical solutions. A patching strategy, based on domain decomposition, is suggested to overcome a deficiency of the LDM. Fractional differential equations are widely investigated by several researchers., College of Arts and Sciences, Department of Mathematics and Statistics

In this paper, a spectral collocation method is proposed and analyzed for solving the time fractional Schrödinger equation. The space derivative is discretized using the collocation method and the time fractional derivative using Grünwald–Letnikov formulation. The stability and convergence of the full discretization scheme are analyzed based on the z-transform. The global behavior of the finite difference spectral collocation method is derived. Numerical examples show a good agreement with the theoretical analysis.

We consider the terminal value problem (or called nal value problem, initial inverse prob- lem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given nal value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of rst and fractional orders are also derived. The e ectiveness of our methods are shown by applying the results to two interesting models: time fractional Ginzburg-Landau equation, and time fractional Burgers equation, where time and spatial regularity estimates are obtained.

We consider the terminal value problem (or called nal value problem, initial inverse prob- lem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given nal value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of rst and fractional orders are also derived. The e ectiveness of our methods are shown by applying the results to two interesting models: time fractional Ginzburg-Landau equation, and time fractional Burgers equation, where time and spatial regularity estimates are obtained.

We consider the terminal value problem (or called nal value problem, initial inverse prob- lem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given nal value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of rst and fractional orders are also derived. The e ectiveness of our methods are shown by applying the results to two interesting models: time fractional Ginzburg-Landau equation, and time fractional Burgers equation, where time and spatial regularity estimates are obtained.

[EN] The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense. Afterwards, we provide reliable explicit approximations for the main statistical information of the solution process (mean, variance and covariance). Further, we show a set of numerical examples where our obtained theory is illustrated. More precisely, we show that our results for the random fractional Airy equation are in full agreement with the corresponding to classical random Airy differential equation available in the extant literature. Finally, we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional Airy differential equation by combining the knowledge of the mean and the variance and the Principle of Maximum Entropy.

[EN] The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense. Afterwards, we provide reliable explicit approximations for the main statistical information of the solution process (mean, variance and covariance). Further, we show a set of numerical examples where our obtained theory is illustrated. More precisely, we show that our results for the random fractional Airy equation are in full agreement with the corresponding to classical random Airy differential equation available in the extant literature. Finally, we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional Airy differential equation by combining the knowledge of the mean and the variance and the Principle of Maximum Entropy.

This thesis mainly concerns the numerical investigation and application of fractional dynamical systems. Two main problems are considered: fractional dynamical models involving the Riesz fractional operator, such as the time-space fractional Bloch-Torrey equation, and complex viscoelastic non-Newtonian Maxwell and Oldroyd-B fluid models. The two main contributions of the research are the treatment of the Riesz space fractional derivative on irregular convex domains and presenting a unified numerical scheme to solve a class of novel multi-term time fractional non-Newtonian fluid models. A rigorous stability and convergence analysis of the computational models is also established.

In this paper, we consider a two-dimensional multi-term time-fractional Oldroyd-B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time-fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L ∞ -norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two-dimensional multi-term time-fractional Oldroyd-B model.

We consider the terminal value problem (or called nal value problem, initial inverse prob- lem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given nal value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of rst and fractional orders are also derived. The e ectiveness of our methods are shown by applying the results to two interesting models: time fractional Ginzburg-Landau equation, and time fractional Burgers equation, where time and spatial regularity estimates are obtained.

In this thesis, we present a computational method for solving a class of fractional singularly perturbed Volterra integro-differential boundary-value problems with a boundary layer at one end. The implemented technique consists of solving two problems which are a reduced problem and a boundary layer correction problem. The reproducing kernel method is used to the second problem. Pade’ approximation technique is used to satisfy the conditions at infinity. Existence and uniformly convergence for the approximate solution are also investigated. Numerical results provided to show the efficiency of the proposed method.

Fractional calculus, the branch of mathematics dealing with derivatives and integrals of non-integer order, began as a mere mathematical curiosity during the time of Leibniz but through the years has developed into a very dynamic field of research. Famous mathematicians such as Riemann, Liouville, Grunwald, Euler, Lagrange, Caputo and others laid the foundation of the modern theory thus paving the way for fractional calculus to enter mainstream mathematics. At present, this field of study is still developing rapidly. New concepts and ideas such as the Caputo-Fabrizio formulation for example, have emerged and applications in such varied fields as viscoelasticity, fluid flow, rheology, etc., have arisen. In this thesis we will address new problems and issues within this area.

In the last three decades, transform methods have been used for solving fractional di erential equations, singular integral equations. In this article, the author considered a new class of the inverse Laplace transforms of exponential types. We also evaluated certain types of integrals and solved partial fractional equations of Cauchy type.The result reveals that the transform method is very convenient and e ective.

In the last three decades, transform methods have been used for solving fractional di erential equations, singular integral equations. In this article, the author considered a new class of the inverse Laplace transforms of exponential types. We also evaluated certain types of integrals and solved partial fractional equations of Cauchy type.The result reveals that the transform method is very convenient and e ective.

In this paper, the homotopy perturbation is successively used to obtain approximate analytical solutions of the nonlinear dispersive $K(2,2)$ equation with time and space derivative. Comparison between the numerical and the exact solutions revealed that HPM is an alternative analytical method for solving fractional differential equations.

In this paper, the homotopy perturbation is successively used to obtain approximate analytical solutions of the nonlinear dispersive $K(2,2)$ equation with time and space derivative. Comparison between the numerical and the exact solutions revealed that HPM is an alternative analytical method for solving fractional differential equations.

we use variational iteration method (VIM) to solve some nonlinear time-fractional advection problem.Compared to the other method, the VIM is direct and straightforward.

We study the uniquenessof solutionfor nonlinear implicit fractional differential equation with initial condition involving Caputo fractional derivative. The technique used in our analysis is based on an application of Bihari and Medved inequalities.

The multi-term time-fractional diffusion equation is a useful tool in the modeling of complex systems. This paper aims to develop a high order numerical method for solving multi-term time-fractional diffusion equations. Based on the space-time spectral method, a high-order scheme is proposed in the present paper. In this method, the Legendre polynomials are adopted in temporal discretization and the Fourier-like basis functions are constructed for the spatial discretization. Such a space-time spectral method possesses high efficiency and exponential decay in both time and space directions. Rigorous proofs are given here for the stability and convergence of the scheme. Numerical results show good agreement with the theoretical analysis.

Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann-Liouville and Hadamard fractional derivatives with order $q \in(0,1)$. Then we obtain exact expression of solutions of impulsive fractional differential equations, i.e., exact expression of piecewise continuous solutions. Finally, four classes of integral type periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. We allow the nonlinearity $p(t) f(t, x)$ in fractional differential equations to be singular at $t=0,1$ and be involved a superlinear and sub-linear term. The analysis relies on Schaefer's fixed point theorem.

Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann-Liouville and Hadamard fractional derivatives with order $q \in(0,1)$. Then we obtain exact expression of solutions of impulsive fractional differential equations, i.e., exact expression of piecewise continuous solutions. Finally, four classes of integral type periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. We allow the nonlinearity $p(t) f(t, x)$ in fractional differential equations to be singular at $t=0,1$ and be involved a superlinear and sub-linear term. The analysis relies on Schaefer's fixed point theorem.

In this paper we deal with the problem of asymptotic integration of a class of fractional differential equations of the Caputo type. The left-hand side of such type of equation is the Caputo derivative of the fractional order r is an element of (n - 1, n) of the solution, and the right-hand side depends not only on ordinary derivatives up to order n - 1 but also on the Caputo derivatives of fractional orders 0 < r(1) < . . . < r(m) < r, and the Riemann-Liouville fractional integrals of positive orders. We give some conditions under which for any global solution x(t) of the equation, there is a constant c is an element of R such that x(t) = ct(R) + o(t(R)) as t --> infinity where R = max{n - 1, r(m)}., V tomto článku je studován problém asymptotické integrace třídy frakcionálních diferenciálních rovnic Caputova typu. Levou stranou těchto rovnic je Caputova derivace řešení neceločíselného řádu r(n-1,n) a pravá strana závisí nejenom na běžných derivacích až do řádu n-1, ale také na Caputových derivacích neceločíselných řádů 0

In this paper we deal with the problem of asymptotic integration of a class of fractional differential equations of the Caputo type. The left-hand side of such type of equation is the Caputo derivative of the fractional order r is an element of (n - 1, n) of the solution, and the right-hand side depends not only on ordinary derivatives up to order n - 1 but also on the Caputo derivatives of fractional orders 0 < r(1) < . . . < r(m) < r, and the Riemann-Liouville fractional integrals of positive orders. We give some conditions under which for any global solution x(t) of the equation, there is a constant c is an element of R such that x(t) = ct(R) + o(t(R)) as t --> infinity where R = max{n - 1, r(m)}., V tomto článku je studován problém asymptotické integrace třídy frakcionálních diferenciálních rovnic Caputova typu. Levou stranou těchto rovnic je Caputova derivace řešení neceločíselného řádu r(n-1,n) a pravá strana závisí nejenom na běžných derivacích až do řádu n-1, ale také na Caputových derivacích neceločíselných řádů 0

In this paper we deal with the problem of asymptotic integration of a class of fractional differential equations of the Caputo type. The left-hand side of such type of equation is the Caputo derivative of the fractional order r is an element of (n - 1, n) of the solution, and the right-hand side depends not only on ordinary derivatives up to order n - 1 but also on the Caputo derivatives of fractional orders 0 < r(1) < . . . < r(m) < r, and the Riemann-Liouville fractional integrals of positive orders. We give some conditions under which for any global solution x(t) of the equation, there is a constant c is an element of R such that x(t) = ct(R) + o(t(R)) as t --> infinity where R = max{n - 1, r(m)}., V tomto článku je studován problém asymptotické integrace třídy frakcionálních diferenciálních rovnic Caputova typu. Levou stranou těchto rovnic je Caputova derivace řešení neceločíselného řádu r(n-1,n) a pravá strana závisí nejenom na běžných derivacích až do řádu n-1, ale také na Caputových derivacích neceločíselných řádů 0

In this thesis, we introduce a new series solutions for multi-term fractional differential equations of Caputo’s type. The idea is similar to the well-known Taylor Series method, but we overcome the difficulty of computing iterated fractional derivatives, which do not commuted in general. To illustrate the efficiency of the new algorithm, we apply it for several types of multi-term fractional differential equations and compare the results with the ones obtained by the well-known Adomian decomposition method (ADM).

In this thesis, we extend the maximum principle and the method of upper and lower solutions to study a class of nonlinear fractional boundary value problems with the Caputo fractional derivative 1

In this thesis a new approach for solving a certain class of anomalous diffusion equations was developed. The theory and algorithms arising from this work will pave the way for more efficient and more accurate solutions of these equations, with applications to science, health and industry. The method of finite volumes was applied to discretise the spatial derivatives, and this was shown to outperform existing methods in several key respects. The stability and convergence of the new method were rigorously established.

In this paper, variational homotopy perturbation method (VHPM) is applied for solving the foam drainage equation with time and space-fractional derivatives. Numerical solutions are obtained for various values of the time and space-order derivative in (0,1]. For the first-order time and space derivative, compared with the exact solution, the result showed that the proposed method could be used as an alternative method for obtaining an analytic and approximate solution for different types of differential equations.

In this paper, variational homotopy perturbation method (VHPM) is applied for solving the foam drainage equation with time and space-fractional derivatives. Numerical solutions are obtained for various values of the time and space-order derivative in (0,1]. For the first-order time and space derivative, compared with the exact solution, the result showed that the proposed method could be used as an alternative method for obtaining an analytic and approximate solution for different types of differential equations.

We establish sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential inclusions involving the Caputo fractional derivative. We consider the cases when the multivalued nonlinear term takes convex values as well as nonconvex values. The topological structure of the set of solutions is also considered.

This paper deals with some existence results for a boundary value problem involving a nonlinear integrodifferential equation of fractional order with integral boundary conditions. Our results are based on contraction mapping principle and Krasnosel'skiĭ's fixed point theorem

Tato práce je zaměřena na popis LTI SISO systémů se zlomkovým řádem pomocí zobecněných Laguerrových funkcí. Jsou zde popsány vlastnosti zobecněných Laguerrových funkcí a je zde graficky zobrazena ortogonální báze těchto funkcí. Dále je vysvětlen pojem zlomkových derivací. Poslední část této práce se zabývá popisem LTI SISO systémů se zlomkovým řádem pomocí zobecněných Laguerrových funkcí. Pro demonstraci vhodnosti použití těchto funkcí k popisu LTI SISO systémů bylo vyřešeno několik příkladů., This paper concentrates on the description of fractional order LTI SISO systems using generalized Laguerre functions. There are properties of generalized Laguerre functions described in the paper, and an orthogonal base of these functions is shown. Next the concept of fractional derivatives is explained. The last part of this paper deals with the representation of fractional order LTI SISO systems using generalized Laguerre functions. Several examples were solved to demonstrate the benefits of using these functions for the representation of LTI SISO systems.

Tato práce je zaměřena na popis LTI SISO systémů se zlomkovým řádem pomocí zobecněných Laguerrových funkcí. Jsou zde popsány vlastnosti zobecněných Laguerrových funkcí a je zde graficky zobrazena ortogonální báze těchto funkcí. Dále je vysvětlen pojem zlomkových derivací. Poslední část této práce se zabývá popisem LTI SISO systémů se zlomkovým řádem pomocí zobecněných Laguerrových funkcí. Pro demonstraci vhodnosti použití těchto funkcí k popisu LTI SISO systémů bylo vyřešeno několik příkladů., This paper concentrates on the description of fractional order LTI SISO systems using generalized Laguerre functions. There are properties of generalized Laguerre functions described in the paper, and an orthogonal base of these functions is shown. Next the concept of fractional derivatives is explained. The last part of this paper deals with the representation of fractional order LTI SISO systems using generalized Laguerre functions. Several examples were solved to demonstrate the benefits of using these functions for the representation of LTI SISO systems.

Tato práce je zaměřena na popis LTI SISO systémů se zlomkovým řádem pomocí zobecněných Laguerrových funkcí. Jsou zde popsány vlastnosti zobecněných Laguerrových funkcí a je zde graficky zobrazena ortogonální báze těchto funkcí. Dále je vysvětlen pojem zlomkových derivací. Poslední část této práce se zabývá popisem LTI SISO systémů se zlomkovým řádem pomocí zobecněných Laguerrových funkcí. Pro demonstraci vhodnosti použití těchto funkcí k popisu LTI SISO systémů bylo vyřešeno několik příkladů., This paper concentrates on the description of fractional order LTI SISO systems using generalized Laguerre functions. There are properties of generalized Laguerre functions described in the paper, and an orthogonal base of these functions is shown. Next the concept of fractional derivatives is explained. The last part of this paper deals with the representation of fractional order LTI SISO systems using generalized Laguerre functions. Several examples were solved to demonstrate the benefits of using these functions for the representation of LTI SISO systems.