This study used the fractional B-spline collocation technique to obtain the numerical solution of fractal-fractional differential equations. The technique was considered to solve the fractal-fractional differential equations (FFDEs) with (). In this suggested technique, the B-spline of fractional order was utilised in the collocation technique. The scheme was easily attained, efficient, and relatively precise with reduced computational work numerical findings. Via the proposed technique, FFDEs can be reduced for solving a system of linear algebraic equations using an appropriate numerical approach. The verified numerical illustrative experiments were presented will show the effectiveness of the technique proposed in this study in solving FFDEs in three cases of nonlocal integral and differential operators namely power law kernel, when the kernels are exponential and the generalization of Mittag-Leffler kernel. The approximate solution is very good and accurate to the exact solution.

In this study, we present the new generalized derivative and integral operators which are based on the newly constructed new generalized Caputo fractal–fractional derivatives (NGCFFDs). Based on these operators, a numerical method is developed to solve the fractal–fractional differential equations (FFDEs). We approximate the solution of the FFDEs as basis vectors of shifted Legendre polynomials (SLPs). We also extend the derivative operational matrix of SLPs to the generalized derivative operational matrix in the sense of NGCFFDs. The efficiency of the developed numerical method is tested by taking various test examples. We also compare the results of our proposed method with the methods existed in the literature In this paper, we specified the fractal–fractional differential operator of new generalized Caputo in three categories: (i) different values in  and fractal parameters, (ii) different values in fractional parameter while fractal and  parameters are fixed, and (iii) different values in fractal parameter controlling fractional and  parameters.

The study of integral equations is one of the most important topics that researchers are interested in, it arises in many scientific fields for instance engineering, and mathematical and scientific analysis. The first who mentioned the term integral equations is Du Bois-Reymond (1888). As a result, a lot of interest appeared from researchers, and the most important of these researchers are Laplace, Fourier, Poission, Liouville, and Able. Upadhyay et al., (2015) provided some special types of integral equations. The quadratic integral equation is a special form of integral equations. The initial study appeared by Chandrasekhar (1947). More appearance of Quadratic integral equation was in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport, and in the traffic theory, see Argyros (1985), Banaś et al. (2007), El-Sayed et al. (2008).

 The aim of the present paper is to develop a generalized fractional kinetic equation involving generalized multi-variable Mittag-Leffler function. Using the Laplace transform, the solutions of the fractional kinetic equation are established in terms on general Mittag-Leffler function. The results obtained here are general in nature to yield a large number known and (presumably) new results as their special cases. 

In this article, an implementation of an efficient numerical method for solving the system of coupled non-linear fractional (Caputo sense) dynamical model of marriage (FDMM) is introduced. The proposed system describes the dynamics of love affair between couples. The method is based on the spectral collocation method using Legendre polynomials. The proposed method reduces FDMM to a system of algebraic equations, which solved using Newton iteration method. Special attention is given to study the convergence analysis and deduce an error upper bound of the resulting approximate solution. Numerical simulation is given to show the validity and the accuracy of the proposed method. 

In this paper we establish the existence of integrable solution of boundary value problems of (mixed type) Fredholm - volterra functional integro-differential equations  with nonlocal boundary conditions.

Mathematical modelling of real-life problems usually results in functional equations, like ordinary or partial differential equations, integral and integro- differential equations, stochastic equations. Many mathematical formulation of physical phenomena contain integro-differential equations, these equations arises in many fields like fluid dynamics, biological models and chemical kinetics integro-differential equations are usually difficult to solve analytically so it is required to obtain an efficient approximate solution.

In this paper we establish the existence of solution for two boundary value problems of Fredholm functional integro-differential equations with nonlocal boundary conditions.