General solution of linear partial differential equations modeling homogeneous diffusion-convection-reaction problems with Cauchy initial condition

In this paper, we propose the general solution of diffusion-convection-reaction homogeneous problems with condition initial of Cauchy, using the SBA numerical method. This method is based on the combination of the Adomian Decompositional Method(ADM), the successive approximations method and the Picard principle. 2010 Mathematics Subject Classifications: 47H14, 34G20, 47J25, 65J15


Introduction
Most physical, medical, biological,...,phenomena are modeled by integral equations, integro-differential equations, ordinary differential equations or by partial differential equations.Generally it is very difficult, or impossible to determine their analytical solutions.In this paper, we propose a general solution of linear homogeneous diffusion, convection and reaction equations with Cauchy initial condition, using the SBA numerical method.

The numerical SBA method
Let us consider the following fonctional equation: Where A : H → H, is an operator not necessarly linear and H is a Hilbert space adequatly chosen given the operator A, f is given function and u the unnown function.
Let : Where L is an invertible operator in the Adomian "sense", R the linear remainder and N a nonlinear operator.The equation (2) therefore becomes : Where θ is such that L (θ) = 0.The equation ( 3) is the Adomian canonical forme , using the successsive approximations [3] we get : This yields the following Adomian algorithm [5] The Picard principle is then applied to the equation ( 5) let u 0 be such that N u 0 = 0, for k = 1, we get : If the series For k = 2, we get: If the series n .This process is repeated to k.

If the series
At each stape k ≥ 1 , we make sure that : N u k = 0.

A convection model
Let us consider the following convection model with Cauchy initial [6][7][8] Appliying the SBA method to (13) at the step k ≥ 0. We obtain the following algorithm : Let us calculate the following terms: We obtain the exact solution of the problem ( 13) Proposition 2. The exact solution of the following convection model with Cauchy initial condition where Proof.Let us consider u (t, x) = ϕ (α (x + λt)).
We obtain : In this case, it is necessary and sufficient that the function ϕ ∈ C 1 (R).Hence u (t, x) = ϕ (α (x + λt)) is the general solution of (16).

A reaction model
Let us consider the following reaction model Cauchy type [6]  Appliying the SBA algorithm to (18) at the step k ≥ 0, we obtain the following algorithm : Let us calculate the following terms: we obtain the exact solution of the problem (18) Proposition 3. The exact solution of the following reaction problem Cauchy type: where Proof.Let us consider u (t, x) = exp (γt) ϕ (αx), we obtain : In this case,it is necessary and sufficient that the function ϕ ∈ C 1 (I) and I ⊂ R or I = R , hence the general solution of (21) is u (t, x) = exp (γt) ϕ (αx) .

A diffusion-convection model
Let us consider the following type of Cauchy linear equation : Applying the algorithm SBA to (D) , we have : Let us determinate the following terms: Then, we obtain In a recurcive way, we obtain : So, the exact solution exact of (D) is Proposition 4. The exact solution of the following reaction problem Cauchy type: where Proof.Let us consider u (t, x) = exp (−εt) ϕ (x + λt) We obtain : In this case, it is necessary and sufficient that the function ϕ ∈ C 1 (I) where I ⊂ R or I = R, hence the general solution of (P 4 ) is u (t, x) = exp (γt) ϕ (αx) .

A reaction model
Proposition 5.The exact solution of the following reaction problem Cauchy type: where Proof.Let us consider u (t, x) = exp (γt) ϕ (αx).We obtain : In this case,it is necessary and sufficient that the function ϕ ∈ C 1 (I) where I ⊂ R or I ⊂ R, hence the general solution of (E) is u (t, x) = exp (γt) ϕ (αx) .

A diffusion-reaction model
Let us consider the following diffision-reaction problem Cauchy type: Appliying the SBA method at the step k ≥ 0, we obtain the following algorithm : (P SBA ) : Let us calculate some terms: We obtain at the same way : we obtain the exact solution of the problem (F ) : Proposition 6.The exact solution of the following diffision-reaction problem Cauchy type where and ϕ verifie the relation : Proof.Let us consider u (t, x) = exp −εα 2 t ϕ (α (x + λt)) .

A diffusion-convection-reaction problem Cauchy type
Let us consider the following diffision-convection-reaction problem Cauchy type: Appliying the SBA method at the step k ≥ 0, we obtain the following algorithm : Let us calculate the following terms: u k 1 (t, x) , u k 2 (t, x) , u k 3 (t, x) , ... Let us consider the following Cauchy linear equation : Applying the algorithm SBA to (P 3 ) , we have :

Conclusion
The SBA numerical method permitted us to resolve a few linear partial differential equations modelling diffusion, convection, reaction problems Cauchy type.The SBA method pemitted us to resolve the problems proposed in this paper.It is then a very powerful numerical tool of analysis for the resolution of these Kinds of problems.