# Rahmat Ali KhanUniversity of Malakand · Department of Mathematics

Rahmat Ali Khan

Ph.D in Mathematics

## About

192

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## Publications

Publications (192)

This manuscript is related to consider a general class of nth order sequential hybrid fractional differential equations (S-HFDEs) with boundary conditions. With the help of the coincidence degree theory of topology, some appropriate results for the existence theory of the aforementioned class are developed. The mentioned degree theory is a powerful...

The goal of this manuscript is to study the existence theory of solution for a nonlinear boundary value problem of tripled system of fractional order hybrid sequential integro-differential equations. The analysis depends on some results from fractional calculus and fixed point theory. As a result, we generalized Darbo's fixed point theorem to form...

In the present paper, we analyze a coupled system of nonlinear three point boundary value problems (BVPs) consisting of a coupled system of higher order hybrid sequential differential equations formulated by fractional operators. We study several new conditions in the direction of existence theory for solutions of the given hybrid system under the...

This paper is concerned about the study of shifted Jacobi polynomials. By means of these polynomials, we construct some operational matrices of fractional order integration and differentiations. Based on these matrices, we develop a numerical scheme for the boundary value problems of fractional order differential equations. The construction of the...

This paper is a study about a fractional-order hybrid system of two fractional-order differential equations. We have given the existence of solution and uniqueness of solutions, and finally, the numerical solutions for an application is presented. The existence results are based on the classical fixed-point approach while the numerical results are...

A wave phenomena evolved day after day, as various concepts regarding waves appeared with the passage of time. These phenomena are generally modelled mathematically by partial differential equations (PDEs). In this research, we investigate the exact analytical solutions of one and two dimensional linear dissipative wave equations which are modelled...

In this paper, we study a class of nonlinear boundary value problems (BVPs) consisting of a more general class of sequential hybrid fractional differential equations (SHFDEs) together with a class of nonlinear boundary conditions at both end points of the domain. The nonlinear functions involved depend explicitly on the fractional derivatives. We s...

Our main goal is to develop some results for transmission of COVID-19 disease through Bats-Hosts-Reservoir-People (BHRP) mathematical model under the Caputo fractional order derivative (CFOD). In first step, derived the feasible region and bounded ness of the model. Also, we derived the disease free equilibrium points (DFE) and basic reproductive n...

In this manuscript, we investigate existence theory as well as stability results to the biological model of HIV (human immunodeficiency virus) disease. We consider the proposed model under Caputo-Fabrizio derivative (CFD) with exponential kernel. We investigate the suggested model from other perspectives by using fixed point approached derive its e...

In this article, we investigate the semi-analytic solutions of non-linear Volterra fractional integro-differential equations by using Laplace Adomian decomposition method. We discuss the method in general and provide examples for the illustration purpose. Moreover, we give existence and uniqueness result for the solutions using Banach contraction p...

This research work is related to investigate sufficient conditions for existence, uniquely of solution to a coupled system of nonlinear fractional order hybrid differential equations (NFOHDEs). With the help of Burton's fixed point theorem, we develop the required conditions for existence of at least one solution to the considered problem of NFOHDE...

In the current manuscript, we investigate existence of solutions to a coupled system of fractional hybrid differential equations (FHDEs). With the help of mixed type Lipschitz and Caratheodory conditions, some conditions for the existence of solutions to the considered problem are established. Considering the tools of nonlinear analysis and hybrid...

Abstract This research work is dedicated to investigating a class of impulsive fractional order differential equations under the Robin boundary conditions via the application of topological degree theory (TDT). We establish some adequate results for the existence of at most one solution for the consider problem. Further, the whole analysis is illus...

In this paper, our main objective is to develop the conditions that assure the existence of solution to a system of boundary value problems (BVPs) of sequential hybrid fractional differential equations (SHFDEs). The problem is considered under the nonlinear boundary conditions. Nonlinear functions involved in the considered system of SHFDEs are con...

In this paper, we use the topological degree theory (TDT) to investigate the existence and uniqueness of solution for a class of evolution fractional order differential equations (FODEs) with proportional delay using Caputo derivative under local conditions. In the same line, we will also study different kinds of Ulam stability such as Ulam–Hyers (...

In this paper, our main purpose is to present an analytical solution for measles spread model with three doses of vaccination using Caputo-Fabrizio fractional derivative (CFFD). The presented solution is based on Laplace transform with Adomian decomposition method (LADM), which is an effective technique to obtain a solution for such type of problem...

With the help of the topological degree theory in this manuscript, we develop qualitative theory for a class of multi-terms fractional order differential equations (FODEs) with proportional delay using the Caputo derivative. In the same line, we will also study various forms of Ulam stability results. To clarify our theoretical analysis, we provide...

In this chapter, we develop an efficient numerical scheme for the solution of boundary value problems of fractional order differential equations as well as their coupled systems by using Bernstein polynomials. On using the mentioned polynomial , we construct operational matrices for both fractional order derivatives and integrations. Also we constr...

In this article we develop series type solution to two dimensional wave equation involving external source term of fractional order. For the require result, we use iterative Laplace transform. The solution is computed in series form which is rapidly convergent to exact value. Some examples are given to illustrate the establish results.

This research is devoted to investigate the existence and multiplicity results of boundary value problem (BVP) for nonlinear fractional order differential equation (FDEs). To obtain the required results, we use some fixed point theorems due to Leggett–Williams and Banach. Further in this paper, we introduce different types of Ulam’s stability conce...

The aims of this manuscript is to establish conditions for obtaining mild solutions to a coupled systems of multipoint boundary value problems (BVPs) of fractional order hybrid differential equations (FHDEs) with nonlinear perturbations of second type. In the concerned problem, we consider a proportional type delay that represent a famous class of...

Abstract In this article, we study the existence result for a boundary value problem (BVP) of hybrid fractional sequential integro-differential equations. A fixed point theorem provided by Dhage in (Nonlinear Anal. 4:414–424, 2010) is used for the solution existence of our boundary value problem. Also we illustrated our result through an example.

Abstract In this article, we consider a study of a general class of nonlinear singular fractional DEs with p-Laplacian for the existence and uniqueness (EU) of a positive solution and the Hyers–Ulam (HU) stability. To proceed, we use classical fixed point theorem and properties of a p-Laplacian operator. The fractional DE is converted into an integ...

In this paper, the first purpose is to study existence and uniqueness of solutions to a system of implicit fractional differential equations (IFDEs) equipped with antiperiodic boundary conditions (BCs). To obtain the mentioned results, we use Schauder's and Banach fixed point theorem. The second purpose is discussing the Ulam-Hyers (UH) and general...

AIDS is one of the major causes of health problems all over the world. In this article the dynamics of Immunology and AIDS model of fractional order is considered. With the help of Laplace transform coupled with the Adomain decomposition method, we develop an analytical scheme to obtain numerical solution for the considered model. The convergent of...

The main goal in this work is to establish a new and efficient analytical scheme for space fractional telegraph equation (FTE) by means of fractional Sumudu decomposition method (SDM). The fractional Available at 782 H. Khan et al. SDM gives us an approximate convergent series solution. The stability of the analytical scheme is also studied. The ap...

In this article, we study the existence and uniqueness of solutions for system of fractional hybrid differential equations.... We established sufficient conditions for the existence and uniqueness of solutions using fixed point theorem on topological degree method. We provide an example to justify the obtained results.

In this manuscript, we are concerned with the existence of solutions to a coupled system of higher order fractional hybrid differential equations (FHDEs). By using the concept of topological degree theory, we establish proper conditions under which the considered coupled system of FHDEs has at least one solution. The respective method is very rarel...

This article is concerned to the study of existence and multiplicity of positive solutions to a class of multi-point boundary value problems of nonlinear fractional order differential equations. Where the nonlinear term is a continuous function. Sufficient conditions for multiplicity results of positive solutions to the problem under consideration...

This paper is devoted to establishing the existence theory for at least one solution to a coupled system of fractional order differential equations (FDEs). The problem under consideration is subjected to movable type integral boundary conditions over a finite time interval. Furthermore, we investigate the approximate solutions to the considered pro...

We study sufficient conditions for existence of solutions to the coupled systems of higher order hybrid fractional differential equations with three-point boundary conditions. For this motive, we apply the coupled fixed point theorem of Krasnoselskii type to form adequate conditions for existence of solutions to the proposed system. We finish the p...

This article is devoted to the study of existence results to a class of boundary value problems for hybrid fractional differential equations. A couple of hybrid fixed point theorems for the sum of three operators are used for proving the main results. Examples illustrating the results are also presented.

The main purpose of this research paper is to prove the existence of solution to the hybrid differential equation of order which satisfied some growth conditions. The concerned results are obtained via using prior estimate method known as topological degree method. In order to prove the existence of fixed point for We prove this by using condensing...

where Dα is the Caputo’s fractional derivative of order α ,1 0 and the functions f : j × R × R → R , f (0,0) = 0 and g : j × R× R → R satisfy certain conditions. The proof of the existence theorem is based on a coupled fixed-point theorem of Krasnoselskii type, which extends a fixed-point theorem of Burton. Finally, our results are illustrated by p...

In this article, we study the existence and uniqueness of solutions for system of fractional hybrid differential equations of order n − 1 < ν ≤ n, without compactness of operator for the given toppled system D ν (x(t) − Θ(t, x(t)) = Φ(t, y(t), I ν y(t)), D ν (y(t) − Θ(t, y(t)) = Φ(t, x(t), I ν x(t)), a.e t ∈ ϑ , n − 1 ≤ ν < n,...

In this manuscript, we use fixed point theorem due to Bashiri theory and develop sufficient conditions for existence of solution of coupled system of fractional differential equation in Banach space.

Where stands for Cupoto fractional derivative of order α where 1< α ≤ 2, J=[0,1], and the functions ƒ : J x R x R → R,ƒ(0,0) and g : J x R x R → R satisfy certain conditions. The proof of the existence theorem is based on a coupled fixed-point theorem of Krasnoselskii type, which extends a fixed-point theorem of Burton. Finally, our results are ill...

In this paper, we study some properties of shifted Legendre Polynomials. Based
on these polynomials, we develop some new operational matrices. These matrices are used
to find approximate solution of fractional order differential equations (FDEs) under boundary
conditions. The idea is then extended to coupled system of FDEs. Some test problems are
s...

In this article, we introduce the triple Laplace transform for the solution of a class of fractional order partial differential equations. As a consequence, fractional order homogeneous heat equation in 2 dimensions is investigated in detail. The corresponding solution is obtained by using the aforementioned triple Laplace transform, which is the g...

This paper is devoted by developing some necessary and sufficient conditions required for the existence of at least one solution to a highly nonlinear toppled system of fractional order boundary value problems with integral boundary conditions. By the use of classical fixed point theorems like Banach contraction theorem, nonlinear alternative of Le...

This article is concerned to the investigation of extremal solutions for a system of fractional order differential equations with coupled integral boundary value problem. In initial stage, we establish a comparison result and then using the iterative technique of monotone type together with the procedure of extremal solutions, we develop sufficient...

In this paper we study the existence of solutions of nonlinear fractional hybrid differential equations. By
using the topological degree theory, some results on the existence of solutions are obtained. The results are
demonstrated by a proper example.

In this manuscript, we investigate some appropriate conditions which ensure
the the existence of at least one solution to a class of non-integer order differential equations(FDEs) provided as
{ z− (t C )|D t=0 qz= (t) = ϕ(z) θ, z (t, z( (1) = t)); δtC∈ DJ pz= [0 (η), p, η , 1], q ∈ ∈ (0 (1,,1) 2]. ,
The nonlinear function θ : J × R → R is a continu...

We established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential
equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows:
𝑐𝐷𝛼[𝑥(𝑡)−𝑓(𝑡, 𝑥(𝑡))] = 𝑔(𝑡, 𝑦(𝑡), 𝐼𝛼𝑦(𝑡)), 𝑡 ∈ 𝐽 = [0, 1], 𝑐𝐷𝛼[𝑦(𝑡)−𝑓(𝑡, 𝑦(𝑡))] = 𝑔(𝑡, 𝑥(𝑡), 𝐼𝛼𝑥...

Due to the increasing application of fractional calculus in engineering and biomedical processes,
we analyze a new method for the numerical simulation of a large class of coupled systems of
fractional-order partial differential equations. In this paper, we study shifted Jacobi polynomials
in the case of two variables and develop some new operationa...

The study of boundary value problems (BVPs) for fractional differential–integral equations (FDIEs) is extremely popular in the scientific community. Scientists are utilizing BVPs for FDIEs in day life problems by the help of different approaches. In this paper, we apply monotone iterative technique for the existence, uniqueness and the error estima...

In the concerned article, we present the numerical solution of nonlinear coupled system
of Whitham-Broer-Kaup equations (WBK) of fractional order. With the help of Laplace transform
coupled with Adomian decomposition method, an iterative procedure is established to investigate
approximate solution to the proposed coupled system of nonlinear partial...

This article is concerned with the study of coupled systems of fractional order hybrid differential equations. We use hybrid fixed point theorem
due to Dhage and develop sufficient conditions for existence of solutions to the
system. We provide an example to demonstrate our main results.

The aim of this paper is developing conditions that guarantee the existence of a solution to a toppled system of differential equations of noninteger order with fractional integral boundary conditions where the nonlinear functions involved in the considered system are continuous and satisfy some growth conditions. We convert the system of different...

We study sufficient conditions for existence and uniqueness of solutions to boundary value problems (BVPs) for fractional hybrid differentialequations(FHDEs) of the form
where I=[0,1],σ,θ ∈(1,2] and δ,ω∈(0,1) . We use hybrid fixed point theorem due to Dhage and develop adequateresults for existence of solutions to the proposed system of (FHDEs). W...

In this paper we studied the existence and uniqueness properties of solution of a fractional order differential equation subject to nonlocal boundary constrains in the form of multi-point boundary conditions. The problems are highly nonlinear fractional order system of differential equations. The system under consideration is a more general form an...

This article is devoted to the study of existence and multiplicity of positive solutions to a class of nonlinear fractional order multi-point boundary value problems of the type −D q 0+ u(t) = f (t, u(t)), 1 < q ≤ 2, 0 < t < 1, u(0) = 0, u(1) = m−2 ∑ i=1 δ i u(η i), where D q 0+ represents standard Riemann-Liouville fractional derivative, δ i , η i...

In this manuscript, we use fixed point theorem due to Bashiri theory and develop sufficient conditions for existence of solution of coupled system of fractional differential equation in Banach space.

This article is concerned with the study of coupled systems of fractional
order hybrid differential equations. We use hybrid fixed point theorem
due to Dhage and develop sufficient conditions for existence of solutions to the
system. We provide an example to demonstrate our main results.

In this article, we study existence, uniqueness and nonexistence of positive solution to a highly nonlinear coupled system of fractional order differential equations. Necessary and sufficient conditions for the existence and uniqueness of positive solution are developed by using Perov’s fixed point theorem for the considered problem. Further, we al...

This paper is devoted to the study of establishing sufficient conditions for existence
and uniqueness of positive solution to a class of non-linear problems of fractional differential equations. The boundary conditions involved Riemann-Liouville fractional
order derivative and integral. Further, the non-linear function f contain fractional
order de...

We study existence and uniqueness of solution to a coupled system of multi-point boundary value problems of highly nonlinear fractional order differential equations. The system under consideration is more general form of coupled system of fractional differential equations and many systems of the aforesaid area are a special cases. By using classica...

In this article, we develop an analytical method for solving fractional order partial differential equations. Our method is the generalizations of Homotopy perturbations Laplace transform method (HPLTM) and Homotopy perturbations Sumudu transform method (HPSTM).
The solutions obtained using the proposed method implies that the method is highly accu...

Optical activity is investigated in a chiral medium by employing the four level cascade atomic model, in which the optical responses of the atomic medium are studied with Kerr nonlinearity. Light entering into a chiral medium splits into circular birefringent beams. The angle of divergence between the circular birefringent beams and the polarizatio...

We investigate sufficient conditions for existence of multiple solutions to a coupled system of fractional-order differential equations with three-point boundary conditions. By coupling the method of upper and lower solutions together with the method of monotone iterative technique, we develop conditions for iterative solutions. Based on these cond...

This paper investigates a computational method to find an approximation to the solution of fractional differential equations subject to local and nonlocal m-point boundary conditions. The method that we will employ is a variant of the spectral method which is based on the normalized Bernstein polynomials and its operational matrices. Operational ma...

In this paper, we have proposed a new formulation for the solution of a general class of fractional differential equations (linear and nonlinear) under ˆ m-point boundary conditions. We derive some new operational matrices and based on these operational matrices we develop scheme to approximate solution of the problem. The scheme convert the bounda...

This paper investigates a computational method to find approximation to the solution of
fractional differential equations subject to local and
nonlocal $m$-point boundary conditions. The method that we will employ is a variant of spectral method which is based on the normalized Bernstein polynomials and its operational matrices. Operational matrice...

In this article, the reproducing kernel Hilbert space 4 2 W [0, 1] is employed for solving a class of third-order periodic boundary value problem by using fitted reproducing kernel algorithm. The reproducing kernel function is built to get fast accurately and efficiently series solutions with easily computable coefficients throughout evolution the...

In this paper we study existence and uniqueness results for a coupled system of nonlinear fractional order differential subject to nonlinear more general four -point boundary condition of the following type where 0 < α, β ≤1 and f, g ∈ C([0, 1] × ℝ2, ℝ) are continuous and the nonlocal functions φ, ψ: (I, ℝ) → ℝ are also continuous. The parameters η...

Enormous application of fractional order partial differential equations (FPDEs) subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world. The prime objective of this article is to find approximate solution of a general FPDEs subject to nonlocal integral type boundary conditi...

In this article, we investigate existence and uniqueness of positive solutions to coupled systems of multi-point boundary value problems for fractional order differential equations of the form
{ D α x ( t ) = ϕ ( t , x ( t ) , y ( t ) ) , t ∈ I = [ 0 , 1 ] , D β y ( t ) = ψ ( t , x ( t ) , y ( t ) ) , t ∈ I = [ 0 , 1 ] , x ( 0 ) = g ( x ) , x ( 1 )...

In this paper we study some interesting properties of shifted Jacobi polynomials and based on these properties a new operational matrix is derived. The new matrix is then used along with some previous results to provide a theoretical treatment to approximate the solution of fractional differential equations with variable coefficients. The scheme is...

This analysis proposes an analytical-numerical approach for providing solutions of a class of nonlinear fractional Klein-Gordon equation subjected to appropriate initial conditions in Caputo sense by using the Fractional Reduced Differential Transform Method (FRDTM). This technique provides the solutions very accurately and efficiently in convergen...

In this paper, we investigate the existence and uniqueness of solution for a non-linear fractional order differential equation (NFDF) c D α u(t)=f(t,u(t)), t∈J=[0, 1] u(o)=g(u), u(1)= δ Γ(q) ∫ o T (t–s) q–1 u(s) ds where 1 < α < 2, and 0 < T, s, q ≤ 1, and f: J × R → R is continuous. In this article we use topological degree method for existence os...

The paper is devoted to the study of operational matrix method for approximating solution of nonlinear coupled system fractional differential equations. The main aim of this article is to approximate solution of the problem under two different types of boundary conditions, $\hat{m}$-point nonlocal boundary conditions and mixed derivative boundary c...

In this paper, we produce numerical solution for a coupled system of partial differential
equations of fractional order (PDEFO) by the help of Bernstein polynomials. This
method reduces the coupled system of PDEFO to a system of algebraic equations which is
simple in handling and gives us good results. The accuracy of the results are examined by
ex...

In this article, we establish a numerical scheme for the numerical solutions of Bagaly Torvik equations. For the numerical
scheme, we construct some operational matrices by using Shifted Legendre polynomials. The considered system is shifted to easily soluble
algebraic equations which is easily computable by using Matlab. We give numerical simula...

We introduce the applications of Natural transform method to find the analytical Solutions of some fractional
order differential equations and also we introduce basic con- cepts of fractional calculus which is new approach to the analytical
solution in this area. First we develop the general setting,then we turn to the applications of Natural tra...

In this paper, we implement a relatively recent analytical technique, called iterative reproducing kernel method (IRKM), to obtain a computational solution for fuzzy two-point boundary value problem based on a generalized differentiability concept. The technique methodology is based on construct a solution in the form of a rabidly convergent series...

This paper deals with the existence of solutions for a class of fractional order differential equations having m-points boundary conditions involving the Caputo fractional derivative. Moreover the nonlinearity also depend on the Caputo fractional derivative. We obtain sufficient conditions for the existence and uniqueness of solutions via Schauder'...

In this paper, we deal with a fractional differential equation of order δ 1 ∈ ( 3 , 4 ] with initial and boundary conditions, D δ 1 ψ ( x ) = − H ( x , ψ ( x ) ) , D α 1 ψ ( 1 ) = 0 = I 3 − δ 1 ψ ( 0 ) = I 4 − δ 1 ψ ( 0 ) , ψ ( 1 ) = Γ ( δ 1 − α 1 ) Γ ( ν 1 ) I δ 1 − α 1 H ( x , ψ ( x ) ) ( 1 ) , where x ∈ [ 0 , 1 ] , α 1 ∈ ( 1 , 2 ] , addressing t...

In this article, we investigate upper and lower solutions to a system of coupled nonlinear fractional differential equations with multi-point coupled boundary conditions. Using monotone type iterative techniques combined with the upper and lower solutions method, some necessary and sufficient conditions are developed to guarantee existence of multi...

We investigate sufficient conditions for existence and uniqueness of solutions for a coupled system of fractional order hybrid differential equations (HDEs) with multi-point hybrid boundary conditions given by
D ω ( x ( t ) H ( t , x ( t ) , z ( t ) ) ) = − K 1 ( t , x ( t ) , z ( t ) ) , ω ∈ ( 2 , 3 ] , D ϵ ( z ( t ) G ( t , x ( t ) , z ( t ) ) )...

In this article, the following fractional order multi-point boundary value problem is considered, where ξ, η,δi,ηi ∈ (0,1) g,h ∈ C(J, ℝ) are given functions and < 1; f: J × ℝ → ℝ is a continuous function and cDq is the Caputo derivative of fractional order q. The notation cDpu(1) means the value of cDpu(t) at t = 1. We use topological degree theory...