New Analytical Methods For Solving Integer And Non-integer Order Models In Finance And Engineering

In this thesis,we introduce some new efficient analytical methods for solving some important fractional(non-integer)and non-fractional(integer)order models in finance,applied physical science and engineering including the non-differentiable problems which arise naturally on the Cantor sets.We discuss the detail procedure of the suggested methods and their convergence analysis.The error estimate of the proposed techniques are also provided.All the proposed analytical methods are successfully validated using some useful models in finance and engineering such as the fractional and non-fractional diffusion equations,the fractional and non-fractional heat equations,the fractional Black-Scholes option pricing equation,the fractional and non-fractional wave equations,the non-differentiable heat,wave and diffusion equations.In Chapter 2,we highlight the basic In Chapter 2,we present a brief history of integer and non-integer order derivatives and integrals in the mathematical literature.Among the fractional derivatives,we briefly discuss the well-known Caputo and Riemann-Liouville fractional derivatives and integrals.Besides,we discuss some recent development on fractional calculus such as the Caputo-Fabrizio and the Atangana-Baleanu fractional derivatives.The Caputo-Fabrizio and the Atangana-Baleanu are new fractional with non-singular kernels.In Chapter 7,we successfully applied the Caputo-Fabrizio and the Atangana-Baleanu fractional derivatives together with a Laplace-type integral transform to fractional Black-Scholes option pricing equation in finance.In Chapter 3,we introduced six new integral transforms namely the J-transform,the Shehu transform,the generalize J?-transform,the fractional J-transform,the fractional J?-transform,and the fuzzy Shehu integral transform.The J-transform is a modification of the Laplace transform,the Sumudu transform,and the natural transform.The Shehu transform is a Laplace-type integral transform which is a generalization of the Laplace transform and the Sumudu integral transform.The fuzzy Shehu transform is defined using using zadeh's decomposition theorem and fuzzy Riemann integrals of real-valued functions on finite intervals and it is a generalization of the well-known fuzzy Laplace transform and the fuzzy Sumudu transform.We provide a brief introduction of fuzzy calculus and fuzzy sets.The real-life applications of the J-transform,the Laplace-type integral transform,and the fuzzy Laplace-type integral transform in finance and engineering are clearly illustrated in Chapter 3,Chapter 6 and Chapter 7 respectively.In Chapter 4,we briefly discuss the basic concept of local fractional calculus fractal media or Cantor sets.We introduced the local fractional natural transform method(LFNTM)and proved its properties.We successfully applied the LFNTM to signal defined on a Cantor sets,non-differentiable Volterra integral equation of the second kind,non-differentiable heat,wave and the diffusion equations.All the applications of the LFNTM are supported with graphical solutions to illustrate the clear behavior of the problems on the Cantor sets.In Chapter 5,we introduced three new analytical methods for solving non-differentiable problems on fractal media.The three analytical methods are the local fractional homotopy analysis method(LFHAM),local fractional natural decomposition method(LFNDM)and the local fractional Laplace homotopy analysis method(LFL-HAM).The LFHAM is a modification of the well-known analytical method called the homotopy analysis method(HAM).We successfully the non-differentiable heat and wave equations using the LFHAM.The local fractional Laplace homotopy analysis method is a combination of the homotopy analysis method and the local fractional Laplace transform method(LFLTM).The LFLHAM is applied to non-differentiable wave equations on fractal media.The LFNDM is coupling of local fractional natural transform method(LFNTM)and the well-known Adomian decomposition method(ADM).We successfully applied the LFNDM to non-differentiable heat equations.We supported all the applications of the three suggested methods with the graphical solutions.In Chapter 6,we proposed three powerful analytical methods for solving integer and non-integer order models arising in engineering and financial industries.The proposed three analytical techniques are:The first analytical method is the homotopy analysis Shehu transform method(HASTM)which is a combination of the well-known technique homotopy analysis method(HAM)which was first proposed by a Chine mathematician(Liao)and a Laplace-like integral transform called the Shehu transform which generalizes the Laplace transform and Sumudu integral transform.The second analytical method is the homotopy analysis fuzzy Shehu transform method(HAFSTM)which is coupling of homotopy analysis method(HAM)and the fuzzy Shehu transform method(FSTM).The FSTM is a generalization of the well-known fuzzy Laplace transform and the fuzzy Sumudu transform.The third analytical method in this chapter is the homotopy perturbation Shehu transform mrthod(HPSTM)which is a combination of homotopy perturbation method which was first devised a Chinese mathematician(He)and Shehu transform method(STM).We clearly proved the error estimate and the convergence of all the proposed analytical methods.All the eficient analytical techniques are successfully applied to many important applications in engineering and finance such as the fractional gas dynamic equation,the fractional diffusion equation,the fractional wave equation and the fuzzy-time fractional partial differential equations.In Chapter 7,introduced two powerful analytical technique for solving non-integer Black-Scholes option pricing equations in finance.We introduced the Natural Homotopy Perturbation Method for Solving European Option Pricing Equation.Proved its convergence analysis and error estimate.The suggested technique proved to be highly efficient and the results obtained are in excellent agreement with the results of the per-existing methods.Secondly,Using a new fractional derivative with singular kernel namely the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives,we proposed a new q-homotopy analysis transform method for solving option pricing equations in finance.We proved the existence and uniqueness of the new option pricing model using the fixed point theorem.Besides,the convergence and error analysis of the proposed iterative technique is proved.The proposed technique converges rapidly and the results obtained axe compared with the results of the existing techniques.